Gaoming Wang

1. Purpose and perspective

Thu, 27 Aug 2026 00:00:00 +0000

The final test function in the paper is compact enough that its origin can be easy to miss. In its finished form it is

\[ \tag{1.1}\label{eq:final-G-intro} G(y)=\left(s_L^2(y)+kP(y)\right)^{1/2}, \qquad P(y)=\prod_{j=1}^m\phi\bigl(1-\ip{y}{p_j}\bigr), \]

where $L=\operatorname{span}\{p_1,\ldots,p_m\}$, $s_L(y)=\abs{\pi_{L^\perp}y}$. Throughout, we call $p_1,\ldots,p_m$ the prescribed directions; they are precisely the zeros of the test function. The term pole is reserved for the multipole construction, its factors, and its numerical configurations. The cutoff $\phi$ is linear near zero and becomes a $2/m$-power away from zero. Read backwards, every part of (1.1) has a reason. The background $s_L^2$, the product structure, the exponent $2/m$, the linear core, the order in which $T$ and $k$ are selected, and the normal-rank restriction were each forced by a different calculation.

The purpose of this note is not only to explain the mathematical origin of this formula, but also to describe how intelligent assistance contributed to the research process. Our principal AI-assisted environments during the original search were a GitHub Copilot subscription and Cursor’s Composer 2. Starting from hand calculations, we used these tools to generate and revise numerical tests quickly, compare alternative implementations of long formulas, scan large parameter families, and concentrate computation in the singular regimes where a proposed test function was most likely to fail.

This assistance was particularly important for numerical validation. It made the cycle from a geometric idea to an executable test sufficiently short that we could examine many more candidates than would otherwise have been practical. A negative numerical value often exposed the exact scale or region in which an ansatz broke down; a positive scan identified a candidate worthy of further analysis. Neither outcome was confused with a proof. The choice of meaningful asymptotic regimes, the interpretation of numerical failures, and the conversion of observed patterns into rigorous estimates remained the mathematical work of the researchers.

This is also a historical account of a rapidly changing technology. In the few months since the original experiments, the intelligence and reasoning ability of leading AI models have improved qualitatively. A comparable search undertaken now may be completed more quickly and with more capable assistance in symbolic reasoning, numerical design, adversarial testing, and proof organization. The workflow described here should therefore not be read as a benchmark for current AI systems. We nevertheless describe it in detail because it gives a concrete example of how AI assistance contributed to this research and of how such tools can help push mathematical understanding further.

Regularity and Compactness Results for Stable Minimal Hypersurfaces

Mon, 29 Jun 2026 00:00:00 +0000

The first result is the generalized Bernstein theorem from Schoen–Simon–Yau.

Theorem 4.1.1

Suppose $M \subset \mathbb{R}^{n+1}$ is a complete, stable, minimal hypersurface without boundary and with at most (intrinsic) Euclidean volume growth. Then, if $n \leq 5$, $M$ must be an affine hyperplane.

Note that the intrinsic Euclidean volume growth condition is weaker than the extrinsic Euclidean volume growth condition. This is also a key condition for the Stable Bernstein Theorem without area growth condition.

Theorem 4.1.2

Let $\{M_k\}$ be a sequence of embedded, stable, orientable minimal hypersurfaces in $B_2^{n+1}(0)$ with the following properties:

$0\in \bar{M}_k$ for each $k$.
$\mathcal{H}^{n-2}(\mathrm{sing}M_k)=0$ for each $k$.
$\mathcal{H}^n(M_k \cap B_2^n(0))\leq \Lambda$ for some constant $\Lambda\gt{}0$ independent of $k$.

Then, up to a subsequence, $M_k$ converges in the varifold sense to a stable minimal hypersurface $M$ in $B_2^{n+1}(0)$, which is smooth except for a closed singular set of Hausdorff dimension at most $n-7$.

In addition, for $n=7$, the singular set is discrete. $M_k$ converges to a stable minimal hypersurface $M$ in the varifold sense means that the varifold $|M_k|$ converges to the varifold $|M|$ in the sense of measures of weak limit.

Recall that a closed set $S$ is of Hausdorff dimension at most $k$ if for any $\varepsilon\gt{}0$, we have $\mathcal{H}^{k+\varepsilon}(S)=0$.

Corollary 4.1.3

Suppose $M \subset \mathbb{R}^{n+1}$ is a complete, stable, embedded minimal hypersurface without boundary and with at most extrinsic Euclidean volume growth. Then, if $n \leq 6$, $M$ must be an affine hyperplane.

The dimension $n\leq 6$ is sharp, as we already proved that Simons’ cone is stable in $\mathbb{R}^8$.

Bellettini’s work completes the corresponding area-growth statement in the borderline dimension $n=6$ for stable immersions. To state the result in a form that is independent of whether one measures volume intrinsically or extrinsically, we also record the comparison theorem of Florit–Simon.

Theorem 4.1.4

Let $\Sigma^d\hookrightarrow \mathbb{R}^N$ be a complete, connected, smooth minimal immersion, and let $p\in \Sigma$. Define the intrinsic and extrinsic area densities by

\[ \mathbf{M}_R^{\mathrm{int}}(\Sigma,p):=\frac{|B_R^\Sigma(p)|}{R^d}, \qquad \mathbf{M}_R^{\mathrm{ext}}(\Sigma,p):=\frac{\mathrm{Area}(\Sigma\cap B_R^{\mathbb{R}^N}(p))}{R^d}, \]

where $B_R^\Sigma(p)=\{x\in \Sigma: d_\Sigma(x,p)\lt{}R\}$, and the extrinsic area is counted with multiplicity. Then $\mathbf{M}_R^{\mathrm{int}}(\Sigma,p)$ and $\mathbf{M}_R^{\mathrm{ext}}(\Sigma,p)$ are monotone nondecreasing in $R$, and

\[ \lim_{R\to\infty}\mathbf{M}_R^{\mathrm{int}}(\Sigma,p) =\lim_{R\to\infty}\mathbf{M}_R^{\mathrm{ext}}(\Sigma,p)\in[\omega_d,\infty], \]

where $\omega_d=|B_1^{\mathbb{R}^d}|$. In particular, $\Sigma$ has bounded intrinsic area density if and only if it has bounded extrinsic area density; in either case, the immersion is proper.

Corollary 4.1.5

Let $2\leq n\leq6$, and let $\Sigma^n\hookrightarrow \mathbb{R}^{n+1}$ be a complete, connected, two-sided, stable minimal immersion. If $\Sigma$ has Euclidean area growth, equivalently

\[ |B_R^\Sigma(p)|\leq \Lambda R^n \quad\text{for some }p\in\Sigma,\ \Lambda\lt{}\infty,\text{ and all }R\gt{}0, \]

\[ \mathrm{Area}(\Sigma\cap B_R^{\mathbb{R}^{n+1}}(p))\leq \Lambda R^n \quad\text{for some }p\in\mathbb{R}^{n+1},\ \Lambda\lt{}\infty,\text{ and all }R\gt{}0, \]

then $\Sigma$ is an affine hyperplane.

For $2\leq n\leq5$, this is the Schoen–Simon–Yau area-growth stable Bernstein theorem. The new borderline case is $n=6$: Bellettini proves the classification under extrinsic Euclidean area growth, and Theorem 4.1.4 converts intrinsic area growth into the same extrinsic hypothesis. Thus the area-growth version of the immersed stable Bernstein theorem is settled in the full sharp range $2\leq n\leq6$.

A natural question is whether the compactness conclusion in Theorem 4.1.2 remains true if we assume $\mathcal{H}^{n-1}(\mathrm{sing}M_k)=0$ instead of $\mathcal{H}^{n-2}(\mathrm{sing}M_k)=0$. The answer is yes, by the following deep result of Wickramasekera.

Theorem 4.1.6

Suppose $V_i$ is a sequence of stationary integral $n$-varifolds in $B_2^{n+1}(0)$ and $V_i$ also satisfies the following conditions:

$0\in \mathrm{spt}\|V_i\|$.
$\|V_i\|(B_2^{n+1}(0))\leq \Lambda$ for some constant $\Lambda\gt{}0$ independent of $i$.
(Stability) Each $V_i$ is stable in $B_2^{n+1}(0)$ on its regular set, i.e., for any $\phi \in C_c^1(\mathrm{reg}V_i)$,

\[ \int_{\mathrm{reg}V_i} |A_i|^2 \phi^2 d\|V_i\| \leq \int_{\mathrm{reg}V_i} |\nabla \phi|^2 d\|V_i\|. \]

(Alpha-Structural Hypothesis) There exists $\alpha\in (0,1)$ such that for each $i$, no point of $\mathrm{spt}\|V_i\|\cap B_1^{n+1}(0)$ has a neighborhood in which $\mathrm{spt}\|V_i\|$ is the union of three or more embedded $C^{1,\alpha}$ hypersurfaces-with-boundary meeting only along their common boundary.

Then, up to a subsequence, $V_i$ converges in the varifold sense to a stationary integral $n$-varifold $V_\infty$ in $B_2^{n+1}(0)$, which is stable and whose singular set in $B_2^{n+1}(0)$ has Hausdorff dimension at most $n-7$.

The theorem is formulated for stationary integral varifolds, so no orientability assumption is part of the statement. In applications to stable immersions, two-sidedness is imposed separately when one writes the stability inequality on the regular set.

The next conjecture concerns compactness for stable minimal immersions with singular sets.

The class of branched two-sided stable minimal $n$-dimensional immersions with the singular set of locally finite $(n-2)$-measure is compact under varifold convergence.

The first result in this direction is the following density-$2$ regularity theorem.

Theorem 4.1.7

Let $\delta \in (0,1)$. Suppose $M_k$ is a sequence of orientable stable minimal hypersurfaces immersed in $B_2^{n+1}(0)$ such that:

$0\in \bar{M}_k$.
$\|M_k\|(B_2^{n+1}(0))\leq (3-\delta)\omega_n 2^n$.
$\mathcal{H}^{n-2}(\mathrm{sing}M_k)=0$.

The general case, with non-optimal singular set dimension, is proved by Hong-Li-Wang [HLW24].

Theorem 4.1.8

Suppose $M_k$ is a sequence of orientable stable minimal hypersurfaces immersed in $B_2^{n+1}(0)$ such that:

$0\in \bar{M}_k$.
$\|M_k\|(B_2^{n+1}(0))\leq \Lambda$.
$\sup_k\operatorname{dim}_{\mathcal{H}}(\mathrm{sing}M_k)\lt{} n-4+\frac{4}{n}$.

Minter–Xiao subsequently proved the optimal non-branched version of this regularity and compactness theorem.

Theorem 4.1.9

Let $n\geq2$. Suppose $M_k$ is a sequence of two-sided stable minimal hypersurfaces smoothly and properly immersed in $B_1^{n+1}(0)$ such that:

$0\in \bar M_k$ for each $k$;
$\sup_k\mathcal H^n(M_k\cap B_1^{n+1}(0))\lt{}\infty$;
$\mathcal H^{n-2}(\mathrm{sing}M_k)=0$ for each $k$, where

\[ \mathrm{sing}M_k:=B_1^{n+1}(0)\cap(\bar M_k\setminus M_k). \]

Then, after passing to a subsequence, $M_k$ converges as varifolds to a stationary integral varifold $V$ in $B_1^{n+1}(0)$. Moreover, there is a relatively closed set

\[ S\subset \mathrm{spt}\,\|V\|\cap B_1^{n+1}(0), \qquad \dim_{\mathcal H}S\leq n-7, \]

such that $V$ is represented on $B_1^{n+1}(0)\setminus S$ by a proper, two-sided, stable minimal immersion, and $M_k$ converges locally smoothly to this immersion away from $S$. In particular, $S=\emptyset$ for $2\leq n\leq6$, while $S$ is discrete for $n=7$.

The hypothesis $\mathcal H^{n-2}(\mathrm{sing}M_k)=0$ is the optimal non-branched assumption. It improves Theorem 4.1.8, where the non-immersed singular set is required to have Hausdorff dimension strictly smaller than $n-4+\frac4n$. The branch point case is not included here: when the non-immersed singular set has positive $\mathcal H^{n-2}$-measure, branch points may occur and the corresponding compactness theory remains a separate problem.

Riemannian Manifolds

Mon, 29 Jun 2026 00:00:00 +0000

Definition 1.1.1

A Riemannian manifold is a smooth manifold $M$ equipped with a Riemannian metric $g$, which is a smooth, positive-definite symmetric $(0,2)$-tensor field.

We use either $g(X,Y)$ or $\left< X,Y \right>$ to denote the inner product of two tangent vectors $X, Y \in TM$ with respect to the metric $g$.

The Riemann tensor $R=R_M$ is defined by

\[ R(X,Y)Z =-\nabla_X \nabla_Y Z + \nabla_Y \nabla_X Z + \nabla_{[X,Y]} Z, \]

Here, $\nabla$ is the Levi-Civita connection associated with $g$, i.e., it satisfies the following properties:

Torsion-free: $\nabla_X Y - \nabla_Y X = [X,Y]$ for all vector fields $X, Y$.
Metric compatibility: $X(g(Y,Z)) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)$ for all vector fields $X, Y, Z$.

The convention for the $(0,4)$-type Riemann curvature tensor is

\[ R(X,Y,Z,W)=g(R(X,Y)Z,W). \]

If the ambient manifold is Euclidean space $\mathbb{R}^N$ with the standard metric, we write $X\cdot Y$ for the inner product of $X$ and $Y$, and $D$ for the standard connection.

Simons’ Identity and Inequality

Mon, 29 Jun 2026 00:00:00 +0000

Lemma 2.1.1

For a minimal hypersurface $\Sigma^n$ in $\mathbb{R}^{n+1}$, the second fundamental form $A$ satisfies the following identity:

\[ \frac{1}{2}\Delta |A|^{2} = -|A|^4 + |\nabla A|^2. \]

Proof. Recall that the Ricci identity states that for any $2$-tensor $T_{ij}$ on $\Sigma$, we have

\[ T_{ij,kl} - T_{ij,lk} = R_{lkim} T_{mj} + R_{lkjm} T_{im}, \]

where $R_{ijkl}$ is the Riemann curvature tensor of $\Sigma$. The Gauss equation in Euclidean space gives

\[ R_{ijkl}=A_{ik}A_{jl}-A_{il}A_{jk}. \]

Hence

\[ \begin{aligned} \frac{1}{2}\Delta |A|^2 ={}& A_{ij}\Delta A_{ij} + A_{ij,k}^{2} \end{aligned} \]

Moreover,

\[ \begin{aligned} A_{ij}\Delta A_{ij} ={}& A_{ij}A_{ij,kk} = A_{ij}A_{ik,jk}\quad (\text{Codazzi equation}) \\ ={}& A_{ij}A_{ik,kj} + A_{ij}(R_{kjim}A_{mk}+R_{kjkm}A_{im}) \quad (\text{Ricci identity}) \\ ={}& A_{ij}A_{kk,ij} + A_{ij}(A_{ki}A_{jm}-A_{ji}A_{km})A_{mk}\\ &+ A_{ij}(A_{kk}A_{jm}-A_{km}A_{jk})A_{im} \quad (\text{Gauss equation, Codazzi equation}) \\ ={}&-A_{ij}A_{ji}A_{km}A_{mk}+ A_{ij}A_{jm}A_{mk}A_{ki}-A_{ij}A_{jk}A_{km}A_{mi}\quad \text{minimal}\\ ={}& -|A|^4. \end{aligned} \]

This proves Simons’ identity. ◻

Lemma 2.1.2

For the second fundamental form $A$ of a minimal hypersurface $\Sigma^n$ in $\mathbb{R}^{n+1}$, we have

\[ |\nabla A|^{2} \geq \left( 1+\frac{2}{n} \right)|\nabla |A||^{2}. \]

Proof. Let $\{e_i\}$ be a local orthonormal frame on $\Sigma$ that diagonalizes $A$ at a point, so $A_{ij} = \lambda_i \delta_{ij}$ with $\sum_i \lambda_i = 0$. Then

\[ |\nabla |A|^2|^2 = \left|\nabla \sum_i \lambda_i^2\right|^2 = \left(2\sum_i \lambda_i A_{ii,k}\right)^2. \]

Hence

\[ \bigl|\nabla|A|^2\bigr|^2 = 4\sum_k\Bigl(\sum_i \lambda_i A_{ii,k}\Bigr)^2 \leq 4 \sum_i \lambda_i^2 \sum_{i,k} A_{ii,k}^2 = 4|A|^2 \sum_{i,k} A_{ii,k}^2. \]

Since $2|A|\,|\nabla|A|| = |\nabla|A|^2|$, we have

\[ |\nabla |A||^2 = \frac{|\nabla|A|^2|^2}{4|A|^2} \leq \sum_{i,k} A_{ii,k}^2. \]

Moreover,

\[ \begin{aligned} \sum_{i,k=1}^{n}A_{ii,k}^2 ={}& \sum_{i\neq k}^{} A_{ii,k}^2 + \sum_{i=1}^{n} A_{ii,i}^2 \\ \leq{}& \sum_{i\neq k}^{} A_{ii,k}^2 + \sum_{i=1}^{n}(\sum_{j\neq i} A_{jj,i})^{2} \\ \leq{}& \sum_{i\neq k}^{} A_{ii,k}^2 + (n-1)\sum_{i=1}^{n}\sum_{j\neq i} A_{jj,i}^2 \\ ={}& n \sum_{i\neq k}^{} A_{ii,k}^2\\ ={}& \frac{n}{2} \sum_{i\neq j,k}^{} A_{ik,i}^{2}+A_{ki,i}^{2} \end{aligned} \]

Therefore,

\[ \left( 1+\frac{2}{n} \right)|\nabla |A||^{2} \leq \sum_{i,k}^{}A_{i i,k}^{2}+\sum_{i\neq j,k}^{} A_{ik,i}^{2}+A_{ki,i}^{2} \leq |\nabla A|^2. \]

◻

Corollary 2.1.3

For a minimal hypersurface $\Sigma^n$ in $\mathbb{R}^{n+1}$, the second fundamental form $A$ satisfies the following differential inequality:

\[ \Delta |A|^{2} \geq -2|A|^4 + 2\left( 1+\frac{2}{n} \right)|\nabla |A||^{2}. \]

Equivalently,

\[ |A|\Delta |A| +|A|^4\geq \frac{2}{n}|\nabla |A||^2. \]

Theorem 2.1.4

Let $\Sigma^n$ be an (immersed) stable minimal hypersurface in $\mathbb{R}^{n+1}$ with $n \leq 5$. Suppose $\Sigma$ has (intrinsic) Euclidean volume growth, i.e., there exists a constant $C\gt{}0$ such that for all $R\gt{}0$,

\[ |B^\Sigma_R| \leq C R^n, \]

where $B^\Sigma_R$ is the intrinsic geodesic ball of radius $R$ in $\Sigma$. Then $\Sigma$ must be a hyperplane.

Proof. We test the stability inequality with $|A|^{p-1}\varphi$. Then

\[ \begin{aligned} \int_{ } |A|^{2p}\varphi^{2} \leq{}& \int_{ } |\nabla (|A|^{p-1}\varphi)|^2 \\ ={}& \int_{ } (p-1)^{2}|A|^{2p-4}|\nabla |A||^2 \varphi^2 + |A|^{2p-2}|\nabla \varphi|^2 + (2p-2)|A|^{2p-3}\varphi \langle\nabla |A|, \nabla \varphi\rangle. \end{aligned} \]

We multiply Simons’ inequality by $|A|^{2p-4}\varphi^2$ and integrate by parts to obtain

\[ \begin{aligned} \frac{2}{n} \int |A|^{2p-4} |\nabla |A||^2 \varphi^2 \leq \int |A|^{2p} \varphi^2-(2p-3)|\nabla |A||^{2}|A|^{2p-4}\varphi^2 - 2|A|^{2p-3}\varphi \langle\nabla |A|, \nabla \varphi\rangle. \end{aligned} \]

Cancelling the $|A|^{2p}\varphi^{2}$ term, we obtain

\[ \left( \frac{2}{n} + (2p-3)-(p-1)^{2} \right) \int |A|^{2p-4} |\nabla |A||^2 \varphi^2 \leq \int |A|^{2p-2}|\nabla \varphi|^2 + (2p-4)|A|^{2p-3}\varphi |\nabla |A|| |\nabla \varphi|. \]

If $\frac{2}{n} + (2p-3)-(p-1)^{2} \gt{} 0$, i.e.,

\[ (p-2)^2\lt{}\frac{2}{n}, \]

then we can apply Cauchy–Schwarz to the right-hand side to get

\[ \int_{ } |A|^{2p-4}|\nabla |A||^2 \varphi^2 \leq C \int_{ } |A|^{2p-2}|\nabla \varphi|^2, \]

for some constant $C$ depending on $n$ and $p$. Using the stability inequality again, we get

\[ \int_{ } |A|^{2p} \varphi^2 \leq C \int_{ } |A|^{2p-2}|\nabla \varphi|^2\leq C \left( \int_{ } |A|^{2p} \varphi^2 \right)^{\frac{p-1}{p}} \left( \int_{ } |\nabla \varphi|^{2p} \right)^{\frac{1}{p}}. \]

Hence,

\[ \int_{ } |A|^{2p} \varphi^2 \leq C \int_{ } |\nabla \varphi|^{2p}. \]

If we choose $\varphi$ to be a cutoff function supported in $B^\Sigma_{2R}$ with $|\nabla \varphi|\leq \frac{C}{R}$, and equal to $1$ in $B^\Sigma_R$, then

\[ \int_{B^\Sigma_R} |A|^{2p} \leq C \int_{B^\Sigma_{2R}} \left( \frac{C}{R} \right)^{2p} \leq C R^{n-2p}. \]

Here the parameter $p$ is half of the final integrability exponent, since the estimate controls $|A|^{2p}$. Thus the decay requires $2p\gt{}n$, equivalently $p\gt{}n/2$. When $n\leq 5$, the admissible interval

\[ 2-\sqrt{\frac{2}{n}}\lt{}p\lt{}2+\sqrt{\frac{2}{n}} \]

intersects $(n/2,\infty)$. Choose such a $p$. Then $n-2p\lt{}0$, and letting $R \to \infty$ gives $A \equiv 0$ on $\Sigma$. Hence $\Sigma$ is a hyperplane. ◻

Another useful hypothesis is extrinsic Euclidean volume growth: there exists $C\gt{}0$ such that for all $R\gt{}0$,

\[ |\Sigma \cap B_R(0)| \leq C R^n. \]

This condition implies intrinsic Euclidean volume growth.

Proposition 2.1.5

The following two statements are equivalent:

Every complete stable minimal hypersurface in $\mathbb{R}^{n+1}$ with extrinsic Euclidean volume growth is a hyperplane. (This means $\Sigma \cap B_R(0)$ has area at most $C R^n$ for some constant $C$ independent of $R$.)
We have the following curvature estimate for stable minimal hypersurfaces: there exists $C=C(n,\Lambda)$ such that if $\Sigma^n \subset B_{2R}(0) \subset \mathbb{R}^{n+1}$ is a stable minimal hypersurface with

\[ |\Sigma \cap B_{2R}(0)| \leq \Lambda R^n, \]

then

\[ \sup_{\Sigma \cap B_R(0)} |A| \leq \frac{C}{R}. \]

Note that (ii) is scale-invariant. If it holds for some $R\gt{}0$, then it holds for all $R\gt{}0$ by scaling. This is because for $\lambda\gt{}0$, the second fundamental form of $\lambda\Sigma$ is $\lambda^{-1}A$, and the area of $\lambda\Sigma\cap B_{2\lambda R}(0)$ is $\lambda^n |\Sigma\cap B_{2R}(0)|$.

Proof. (ii) $\Rightarrow$ (i). Let $\Sigma^n\subset \mathbb{R}^{n+1}$ be complete, stable, and minimal, with extrinsic Euclidean volume growth:

\[ |\Sigma\cap B_R(0)|\leq C_0 R^n \quad \forall R\gt{}0. \]

Fix $p\in\Sigma$. For $R\geq |p|$, the ball $B_{2R}(p)$ is contained in $B_{3R}(0)$, and hence

\[ |\Sigma\cap B_{2R}(p)|\leq C_0(3R)^n. \]

Apply (ii) after translating $p$ to the origin, with $\Lambda=3^nC_0$:

\[ \sup_{\Sigma\cap B_R(p)}|A|\leq \frac{C}{R}. \]

Let $R\to\infty$. Then $|A|(p)=0$. Since $p$ is arbitrary, $A\equiv 0$, so $\Sigma$ is a hyperplane.

(i) $\Rightarrow$ (ii). Assume (ii) fails. Then for some $\Lambda\gt{}0$ there exist stable minimal hypersurfaces

\[ \Sigma_j\subset B_{2R_j}(0),\qquad |\Sigma_j\cap B_{2R_j}(0)|\leq \Lambda R_j^n, \]

such that

\[ \sup_{\Sigma_j\cap B_{R_j}(0)} |A_j|\,R_j \to \infty. \]

Choose $y_j\in\Sigma_j\cap B_{R_j}(0)$ such that, with $K_j:=|A_j|(y_j)$,

\[ K_jR_j\to\infty. \]

We use the standard point-picking argument in the ball centered at $y_j$. Define

\[ G_j(x):=\left(\frac{R_j}{2}-|x-y_j|\right)|A_j|(x) \]

on $\Sigma_j\cap B_{R_j/2}(y_j)$, and choose $q_j$ where $G_j$ attains its maximum. Since $G_j(y_j)=R_jK_j/2$, if

\[ Q_j:=|A_j|(q_j),\qquad \rho_j:=\frac{1}{2}\left(\frac{R_j}{2}-|q_j-y_j|\right), \]

then

\[ Q_j\rho_j=\frac{G_j(q_j)}{2}\geq \frac{R_jK_j}{4}\to\infty. \]

Moreover, for $x\in\Sigma_j\cap B_{\rho_j}(q_j)$,

\[ \frac{R_j}{2}-|x-y_j| \geq \frac{1}{2}\left(\frac{R_j}{2}-|q_j-y_j|\right), \]

and the maximality of $G_j$ gives

\[ |A_j|(x)\leq 2Q_j. \]

Rescale:

\[ \widetilde\Sigma_j:=Q_j(\Sigma_j-q_j). \]

Then $0\in\widetilde\Sigma_j$, $|\widetilde A_j|(0)=1$, and on $B_{\widetilde\rho_j}(0)$ with $\widetilde\rho_j:=Q_j\rho_j\to\infty$:

\[ |\widetilde A_j|\leq 2. \]

Stability is scale-invariant, so each $\widetilde\Sigma_j$ is stable minimal.

We now get the local area bound in the rescaled sequence. Because $q_j\in B_{R_j/2}(y_j)$ and $y_j\in B_{R_j}(0)$, we have $q_j\in B_{3R_j/2}(0)$. Hence

\[ B_{R_j/2}(q_j)\subset B_{2R_j}(0). \]

Fix $\sigma\gt{}0$. Since $\rho_j\leq R_j/4$ and $Q_j\rho_j\to\infty$, we also have $Q_jR_j\to\infty$. For $j$ large, $\sigma/Q_j\leq \rho_j$ and $\sigma/Q_j\leq R_j/2$. By the monotonicity formula, applied with center $q_j$ and the two radii $\sigma/Q_j$ and $R_j/2$,

\[ \frac{|\Sigma_j\cap B_{\sigma/Q_j}(q_j)|}{(\sigma/Q_j)^n} \leq \frac{|\Sigma_j\cap B_{R_j/2}(q_j)|}{(R_j/2)^n} \leq 2^n\Lambda. \]

Hence

\[ |\widetilde\Sigma_j\cap B_\sigma(0)| =Q_j^n |\Sigma_j\cap B_{\sigma/Q_j}(q_j)| \leq 2^n\Lambda \sigma^n. \]

Thus $\widetilde\Sigma_j$ have uniform local area growth and curvature bounds on larger and larger balls. By the compactness theorem for minimal immersions with locally bounded curvature and area, a subsequence converges smoothly on compact sets to a complete stable minimal hypersurface $\Sigma_\infty\subset\mathbb{R}^{n+1}$ with Euclidean volume growth. By construction,

\[ |A_{\Sigma_\infty}|(0)=1. \]

But (i) says every such $\Sigma_\infty$ is a hyperplane, so $A_{\Sigma_\infty}\equiv 0$, contradiction.

Therefore (ii) must hold. ◻

Corollary 2.1.6

The curvature estimate in (ii) holds for $n\leq 5$.

Theorem 2.1.7

Let $u:\mathbb{R}^n\to \mathbb{R}$ be an entire solution of the minimal surface equation, with $n\leq 5$. Then $u$ is affine.

Proof. Let

\[ \Sigma=\{(x,u(x)):x\in \mathbb{R}^n\}\subset \mathbb{R}^{n+1} \]

be the graph of $u$. By the calibration argument above, $\Sigma$ is area-minimizing. In particular, $\Sigma$ is stable.

We next prove Euclidean volume growth. After a translation, we may assume $0\in \Sigma$. For a.e. $R\gt{}0$, the intersection

\[ \Gamma_R:=\Sigma\cap \partial B_R \]

is a smooth closed $(n-1)$-dimensional submanifold of the sphere $\partial B_R=S_R^n$. Since

\[ H_{n-1}(S^n)=0, \]

the cycle $\Gamma_R$ bounds an $n$-dimensional region $D_R\subset S_R^n$. Choosing the smaller of the two sides of $S_R^n\setminus \Gamma_R$, we may assume

\[ |D_R|\leq \frac12 |S_R^n|\leq C_n R^n. \]

Since $\Sigma$ is area-minimizing and

\[ \partial(\Sigma\cap B_R)=\Gamma_R=\partial D_R, \]

we obtain

\[ |\Sigma\cap B_R|\leq |D_R|\leq C_n R^n \]

for a.e. $R\gt{}0$.

Hence, by the Schoen–Simon–Yau theorem, $\Sigma$ is a hyperplane. Since $\Sigma$ is a graph over $\mathbb{R}^n$, that hyperplane cannot be vertical. Thus

\[ u(x_1,\ldots,x_n)=a_1 x_1 + \cdots + a_n x_n + c \]

◻

Stable Bernstein in Dimension Two

Mon, 29 Jun 2026 00:00:00 +0000

Theorem 5.1.1

Let $M$ be a complete, stable minimal surface in $\mathbb{R}^3$. Then $M$ is a plane.

The proof presented here is based on the method of Fischer–Colbrie and Schoen [FCS80]. The argument has three main ingredients:

Properties of the Schrödinger operator $-\Delta+q$, or equivalently of the equation $(\Delta-q)g=0$, on complete Riemannian manifolds.
Properties of the operator $\Delta-a K$ on conformal metrics on the disc, where $K$ is the Gauss curvature function and $a$ is a constant.
Classification of the topology of stable minimal surfaces in 3-manifolds with non-negative scalar curvature.

Properties of differential operators on complete Riemannian manifolds

Let $(M, g)$ be a complete $n$-dimensional Riemannian manifold, and let $q$ be a smooth function on $M$. For any bounded domain $D \subset M$ with smooth boundary, we denote by $\lambda_{1}(D)\lt{}\lambda_{2}(D) \leq \lambda_{3}(D) \leq \cdots$ the Dirichlet eigenvalues of the Schrödinger operator $-\Delta+q$ on $D$, where $\Delta$ is the Laplace–Beltrami operator with respect to the metric $g$. Thus $\lambda_1(D)$ is the bottom of the quadratic form associated with the equation $(\Delta-q)g=0$. The standard variational characterization of the first eigenvalue is given by

\begin{equation} \tag{5.1.1} \label{eq:variational} \lambda_{1}(D)=\inf \left\{\int_{D}\left(|\nabla f|_{g}^{2}+q f^{2}\right) \mathrm{dvol}_{g}: \mathrm{spt}\, f \subset D, \int_{D} f^{2} \mathrm{dvol}_{g}=1\right\}, \end{equation}

where $|\nabla f|_{g}$ denotes the norm of the gradient of $f$ with respect to the metric $g$, and $\mathrm{dvol}_{g}$ is the volume form induced by $g$. The following is a fundamental property:

Lemma 5.1.2

If $D, D'$ are connected domains in $M$ with $D \subset D'$, then $\lambda_{1}(D) \geq \lambda_{1}(D')$. Moreover, if $D' \setminus \bar{D} \neq \emptyset$, then $\lambda_{1}(D)\gt{}\lambda_{1}(D')$.

We now state the main result of this section.

Theorem 5.1.3

The following conditions are equivalent:

$\lambda_{1}(D) \geq 0$ for every bounded domain $D \subset M$;
$\lambda_{1}(D)\gt{}0$ for every bounded domain $D \subset M$;
there exists a positive function $g$ satisfying the equation $\Delta g-q g=0$ on $M$.

Proof. (i) $\Rightarrow$ (ii). This is a consequence of Lemma 5.1.2 since, for any bounded domain $D \subset M$ and any point $x_{0} \in M$ we can choose $R$ large enough so that the ball $B_{R}(x_{0})=\{x \in M: \mathrm{dist}(x, x_{0})\lt{}R\}$ satisfies $B_{R}(x_{0}) \setminus \bar{D} \neq \emptyset$ and we have $\lambda_{1}(B_{R}(x_{0})) \geq 0$ by hypothesis.

$ii$ $\Rightarrow$ (iii). To prove the existence of a positive solution $g$ of $\Delta g-q g=0$ we fix a point $x_{0} \in M$. For each $R\gt{}0$ we consider the problem

\[ \begin{cases} \Delta u-q u=0 & \text{on } B_{R}(x_{0}), \\ u=1 & \text{on } \partial B_{R}(x_{0}). \end{cases} \]

Since $\lambda_{1}(B_{R}(x_{0}))\gt{}0$, the Fredholm alternative thus implies the existence of the above problem.

We now prove that $u\gt{}0$ on $B_{R}(x_{0})$. It follows from the strong maximum principle that if $u \geq 0$ on $B_{R}(x_{0})$, then $u\gt{}0$ on $B_{R}(x_{0})$. Suppose now that $\Omega=\{x \in B_{R}(x_{0}): u(x)\lt{}0\} \neq \emptyset$. Hence $\Omega \subset B_{R}(x_{0})$ is a bounded domain and thus, by Lemma 5.1.2, $\lambda_{1}(\Omega)\gt{}0$. Since $\Delta u-q u=0$ on $\Omega$ and $u=0$ on $\partial \Omega$, we would have $u \equiv 0$ in $\Omega$ contradicting the unique continuation property. We have shown that $u\gt{}0$ on $B_{R}(x_{0})$.

We now set $g_{R}(x)=u(x_{0})^{-1} u(x)$ for $x \in M$. We have seen that $g_{R}$ satisfies

\[ \Delta g_{R}-q g_{R}=0 \text{ on } B_{R}(x_{0}), \quad g_{R}(x_{0})=1, \, g_{R}\gt{}0 \text{ on } B_{R}(x_{0}). \]

From the Harnack inequality, it follows that on any ball $B_{\sigma}(x_{0})$, there is a constant $C$ depending only on $\sigma$ and $M$ (independent of $R$) such that, for $R\gt{}2\sigma$

\[ g_{R} \leq C \text{ on } B_{\sigma}(x_{0}). \]

It now follows from standard elliptic theory that all derivatives of $g_{R}$ are bounded uniformly (independent of $R$) on compact subsets of $M$. We may therefore choose a sequence $R_{i} \to \infty$ so that $g_{R_{i}}$ converges along with its derivatives on any compact subset of $M$, and by taking a diagonal sequence we can arrange that $g_{R_{i}}$ along with its derivatives, converges uniformly on compact subsets of $M$ to a function $g$ satisfying $\Delta g-q g=0$ and $g(x_{0})=1$. Since $g$ is not identically zero and $g \geq 0$ the strict maximum principle implies that $g\gt{}0$. This finishes the proof that (ii) $\Rightarrow$ (iii).

$iii$ $\Rightarrow$ (i). If $g\gt{}0$ satisfies $\Delta g-q g=0$ on $M$ we define a new function $w=\log g$. We now calculate

\begin{equation} \tag{5.1.2} \label{eq:w} \Delta w=q-|\nabla w|^{2}. \end{equation}

Let $f$ be any function with compact support on $M$. Multiplying (5.1.2) by $f^{2}$ and integrating by parts, we obtain

\[ -\int_{M} q f^{2} \,\mathrm{dvol}+\int_{M}|\nabla w|^{2} f^{2} \,\mathrm{dvol}=2 \int_{M} f\langle \nabla f, \nabla w\rangle \,\mathrm{dvol}. \]

Applying the Schwarz inequality and the arithmetic-geometric mean inequality we have

\[ 2|f\langle \nabla f, \nabla w\rangle| \leq 2|f||\nabla f||\nabla w| \leq f^{2}|\nabla w|^{2}+|\nabla f|^{2}. \]

Putting this into the above equation and canceling the terms $\int_{M} f^{2}|\nabla w|^{2}$ we obtain

\[ -\int_{M} q f^{2} \,\mathrm{dvol}\leq \int_{M}|\nabla f|^{2} \,\mathrm{dvol}. \]

If $D$ is any bounded domain and $f$ is any function with support in $D$ we have shown that

\[ \int_{D}\left(|\nabla f|^{2}+q f^{2}\right) \,\mathrm{dvol}\geq 0. \]

It now follows from (5.1.1) that $\lambda_{1}(D) \geq 0$. This finishes the proof of Theorem 5.1.3. ◻

The last part of the proof actually yields

Corollary 5.1.4

If $D \subset M$ is any bounded domain, and if there is a function $g\gt{}0$ in $D$ satisfying $\Delta g-q g=0$, then $\lambda_{1}(D) \geq 0$.

The Operator $\Delta-aK$ on Surfaces

Let $M$ be the unit disc in the complex plane endowed with the metric $\mathrm{d}s^{2}=\mu(z)|d z|^{2}$. We assume $\mathrm{d}s^{2}$ is a complete metric. Let $K$ denote the Gaussian curvature of $M$ and $\Delta$ the metric Laplacian, i.e., $\Delta f=\mu^{-1}(f_{xx}+f_{yy})$, where $z=x+i y$. The well-known formula for $K$ is $K=-\frac{1}{2} \Delta \log \mu$. We shall prove the following theorem.

Theorem 5.1.5

Assume $\mathrm{d}s^{2}$ is complete. For $a \gt{} \frac{1}{2}$ there is no positive solution $g$ of $\Delta g-a K g=0$ on $M$.

This is the key analytic input for determining the conformal type of stable minimal surfaces in $\mathbb{R}^3$. Fischer–Colbrie and Schoen [FCS80] prove the case $a=1$, while the method of do Carmo and Peng [dCP79] gives the stated range $a\gt{}\frac{1}{2}$. For the Poincaré metric on the disc the critical value is $\frac{1}{4}$. Under the additional assumption $K\leq 0$, Kawai [Kaw88] proves the corresponding nonexistence result for $a\gt{}\frac{1}{4}$.

Proof. We define a function $h$ by $h=\mu^{-1/2}$. We see from the definition of $K$ that $\Delta \log h=K$, i.e.,

\[ \frac{\Delta h}{h}-\frac{|\nabla h|^{2}}{h^{2}}=K. \]

In particular, $h$ satisfies

\begin{equation} \tag{5.1.3} \label{eq:h} h\Delta h=K h^2+|\nabla h|^{2}. \end{equation}

Let $D \subset M$ be a bounded domain, and let $\zeta$ be a smooth function on $M$ with compact support in $D$. We now calculate

\begin{equation} \tag{5.1.4} \label{eq:calc1} \begin{aligned} \int_{M}\left(|\nabla(\zeta h)|^{2}+aK(\zeta h)^{2}\right) &=\int_{M} |\nabla\zeta|^{2} h^{2} + \frac{1}{2}\langle \nabla\zeta^2, \nabla h^2\rangle + \zeta^2 |\nabla h|^2+aK(\zeta h)^{2}\\ &=\int_{M}|\nabla\zeta|^{2} h^{2}+\frac{1-a}{2}\langle \nabla\zeta^2, \nabla h^2\rangle+\frac{1-a}{2}\zeta^2|\nabla h|^2\\ &\quad - \frac{a}{2}\zeta^2 h \Delta h +aK(\zeta h)^{2} \,\mathrm{dvol}\\ &=\int_{M}\left(|\nabla\zeta|^{2} h^{2} +\frac{1-2a}{2} \zeta^2 |\nabla h|^{2}\right) + \frac{1-a}{2} \langle \nabla\zeta^2, \nabla h^2\rangle\\ &\leq \int_{M}C|\nabla\zeta|^{2} h^{2} -\varepsilon\int_{M}|\nabla h|^{2} \zeta^{2} \,\mathrm{dvol}. \end{aligned} \end{equation}

So we have

\begin{equation} \tag{5.1.5} \label{eq:calc3} \lambda_{1}(D) \int_{M}(\zeta h)^{2} \,\mathrm{dvol}\leq \int_{M}|\nabla\zeta|^{2} h^{2} \,\mathrm{dvol}-\int_{M}|\nabla h|^{2} \zeta^{2} \,\mathrm{dvol}. \end{equation}

Now define a smooth function $\zeta(r)$ for $r \in \mathbb{R}$ which satisfies

\begin{equation} \tag{5.1.6} \label{eq:zeta} \begin{gathered} \zeta(r)=1 \text{ for } r \leq \frac{1}{2} R, \, \zeta(r)=0 \text{ for } r \geq R, \\ \zeta \geq 0 \text{ for all } r, \quad |\zeta'| \leq \frac{3}{R} \text{ for all } r. \end{gathered} \end{equation}

If $r$ measures the metric distance to $0$, and $R$ is any positive number, then $\zeta(r)$ defines a Lipschitz function on $M$ with support in $B_{R}(0)$. A standard approximation argument justifies this choice of $\zeta$ in (5.1.5). Then

\[ \int_{M}|\nabla\zeta|^{2} h^{2} \,\mathrm{dvol}\leq \frac{9}{R^{2}} \int_{M} \mathrm{d}x \mathrm{d}y=\frac{9\pi}{R^{2}}. \]

Putting this into (5.1.5) we have

\begin{equation} \tag{5.1.7} \label{eq:calc4} \lambda_{1}\left(B_{R}(0)\right) \int_{M}(\zeta h)^{2} \,\mathrm{dvol}\leq \frac{9\pi}{R^{2}}-\int_{M}|\nabla h|^{2} \zeta^{2} \,\mathrm{dvol}. \end{equation}

Since $\mu(z)|d z|^{2}$ is a complete metric on the disc, $\mu$ cannot be a constant function. Therefore, $|\nabla h|^{2}$ is not identically zero on $M$. Thus, by choosing $R$ sufficiently large in (5.1.7), we conclude that $\lambda_{1}(B_{R}(0))\lt{}0$. By Theorem 5.1.3 this implies that there is no positive solution of $\Delta g-aKg=0$ on $M$. This completes the proof of Theorem 5.1.5. ◻

The next result is an extension of Theorem 5.1.5 with an additional non-negative potential. It follows directly from Theorem 5.1.3, Theorem 5.1.5, and formula (5.1.1).

Corollary 5.1.6

Let $\mathrm{d}s^{2}=\mu(z)|d z|^{2}$ be a complete metric on the disc. If $a \geq 1$ and $P$ is a non-negative function, then there is no positive solution $g$ of $\Delta g-a K g+P g=0$ on $M$.

Complete Stable Minimal Surfaces in 3-Manifolds

The stability of $M$ is given by the following inequality:

\begin{equation} \tag{5.1.8} \label{eq:stability1} \int_{M}\left[|\nabla f|^{2}-\left(\mathrm{Ric}(e_{3})+\sum_{i,j=1}^{2} h_{ij}^{2}\right) f^{2}\right] \,\mathrm{dvol}\geq 0, \end{equation}

where $f$ is any function having compact support on $M$ and $\mathrm{Ric}(e_{3})$ is the Ricci curvature of $N$ in the direction of $e_{3}$. We now do the rearrangement described in Schoen–Yau [SY79a]. The Gauss curvature equation says that

\[ K=K_{12}+h_{11}h_{22}-h_{12}^{2}, \]

where $K$ is the intrinsic Gaussian curvature of $M$ and $K_{ij}$ is the sectional curvature of $N$ for the section determined by $e_{i}$ and $e_{j}$.

Using minimality and symmetry of $h_{ij}$ we have

\[ K=K_{12}-\frac{1}{2} \sum_{i,j=1}^{2} h_{ij}^{2}. \]

Inequality (5.1.8) may then be written in the form

\begin{equation} \tag{5.1.9} \label{eq:stability2} \int_{M}\left[|\nabla f|^{2}-\left(S-K+\frac{1}{2} \sum_{i,j=1}^{2} h_{ij}^{2}\right) f^{2}\right] \,\mathrm{dvol}\geq 0, \end{equation}

where $S$ is the scalar curvature of $N$ given by $S=2(K_{12}+K_{13}+K_{23})$.

Set

\[ P:=S-K+\frac{1}{2} \sum_{i,j=1}^{2} h_{ij}^{2}. \]

According to (5.1.1), this inequality is equivalent to

\[ \lambda_1(D;-\Delta-P)\geq0 \]

for every bounded domain $D \subset M$, i.e. to Theorem 5.1.3 with $q=-P$. The associated equation is given by the stability operator

\begin{equation} \tag{5.1.10} \label{eq:operator} \Delta+\left(S-K+\frac{1}{2} \sum_{i,j=1}^{2} h_{ij}^{2}\right). \end{equation}

We now classify the stable minimal surfaces in three-manifolds of non-negative scalar curvature.

Theorem 5.1.7

Let $N$ be a complete oriented 3-manifold of non-negative scalar curvature. Let $M$ be an oriented complete stable minimal surface in $N$. There are two possibilities:

If $M$ is compact, then $M$ is conformally equivalent to the sphere $S^{2}$ or $M$ is a totally geodesic flat torus $T^{2}$. If $S\gt{}0$ on $N$, then $M$ is conformally equivalent to $S^{2}$.
If $M$ is not compact, then $M$ is conformally equivalent to the complex plane $\mathbb{C}$ or the cylinder $\Lambda$. If $M$ is a cylinder and the absolute total curvature of $M$ is finite, then $M$ is flat and totally geodesic. If the scalar curvature of $N$ is everywhere positive, then $M$ cannot be a cylinder with finite total curvature.

If the Ricci curvature of $N$ is non-negative, then the assumption of finite total curvature in (ii) can be removed.

Before giving the proof of Theorem 5.1.7 we state the following corollary for the case when $N$ is $\mathbb{R}^{3}$. This implies the classical Bernstein theorem [Ber17] for complete minimal graphs in $\mathbb{R}^{3}$.

Corollary 5.1.8

The only complete stable oriented minimal surface in $\mathbb{R}^{3}$ is the plane.

Proof. In this case the stability operator (5.1.10) becomes $\Delta-2K$ and by Theorem 5.1.7 we know that $M$ is conformally either $\mathbb{C}$ or $\Lambda$. By Theorem 5.1.3 there is a positive function $g$ on $M$ satisfying $\Delta g-2K g=0$. If $M$ is conformal to $\Lambda$ we may lift $g$ to the universal covering $\mathbb{C}$ of $\Lambda$. In either case we have a metric on $\mathbb{C}$ with $K \leq 0$ and a positive $g$ satisfying $\Delta g-2K g=0$. Thus $\Delta g \leq 0$, and $g$ is a positive superharmonic function on $\mathbb{C}$ which must be constant. Therefore $K$ is identically zero and hence $\sum_{i,j} h_{ij}^{2}=-2K$ is identically zero. Consequently $M$ is a plane. ◻

Observe that each of the four possibilities of Theorem 5.1.7 does occur. For example, $S^{2} \times \mathbb{R}$ has positive scalar curvature and has a stable $S^{2}$, $T^{2} \times \mathbb{R}$ is flat and has a stable $T^{2}$. We can choose a metric on $\mathbb{C}$ of positive Gaussian curvature and by crossing with $\mathbb{R}$ construct a metric of positive scalar curvature on $\mathbb{R}^{3}$ having a stable $\mathbb{C}$. Similarly $\Lambda \times \mathbb{R}$ has a flat metric with a stable $\Lambda$.

Proof. Case (i) was observed by Schoen–Yau [SY79a] and follows by choosing $f$ identically equal to one in inequality (5.1.9) to obtain

\[ \int_{M}\left(S+\frac{1}{2} \sum h_{ij}^{2}\right) \,\mathrm{dvol}\leq \int_{M} K \,\mathrm{dvol}. \]

The Gauss-Bonnet theorem now implies that $M$ is the sphere or the torus. In the torus case $S \equiv 0$ on $M$. The stability operator reduces to $\Delta-K$ and its first eigenvalue is

\[ \lambda_{1} \equiv \inf \left\{\int_{M}\left(|\nabla f|^{2}+K f^{2}\right) \,\mathrm{dvol}: \int_{M} f^{2}=1\right\}. \]

Since $\lambda_{1} \geq 0$ by stability and $\int_{M} K \,\mathrm{dvol}=0$ we conclude that $\lambda_{1}=0$ and the constant function $f \equiv 1$ satisfies $\Delta f-K f=0$ which implies that $K \equiv 0$.

To prove case (ii), we first show that the universal covering of $M$ is conformally equivalent to $\mathbb{C}$. If this is not true, then $M$ is covered by the disc. Using stability and Theorem 5.1.3 we have a positive function $g$ on $M$ satisfying

\[ \Delta g-K g+\left(S+\frac{1}{2} \sum_{i,j=1}^{2} h_{ij}^{2}\right) g=0. \]

Lifting $g$ to the disc we obtain a positive solution of this equation on the disc endowed with a complete metric. Since $S+\frac{1}{2} \sum_{i,j=1}^{2} h_{ij}^{2} \geq 0$, this yields a contradiction by Corollary 5.1.6. Thus we have shown that $M$ is conformally covered by $\mathbb{C}$ and hence $M$ is either conformally equivalent to $\mathbb{C}$ or $M$ is conformal to a cylinder $\Lambda$.

If $M$ is a cylinder, let $z=x+i y$ be a complex coordinate for $M$ so that $|d z|^{2}$ is the flat metric on $M$, and the given metric on $M$ is $\mathrm{d}s^{2}=\mu(z)|d z|^{2}$. Fix a point $z_{0} \in M$ and let $r$ be the distance from $z_{0}$ taken with respect to the flat metric. For any $R\gt{}0$, choose $\zeta(r)$ satisfying (5.1.6). Substituting $\zeta$ for $f$ in (5.1.9) and using (5.1.6) and the conformal invariance of the Dirichlet integral we have

\[ \frac{9}{R^{2}} \int_{B_{R}(z_{0})} \mathrm{d}x \mathrm{d}y-\int_{M}\left(S-K+\frac{1}{2} \sum h_{ij}^{2}\right) f^{2} \,\mathrm{dvol}\geq 0, \]

where $B_{R}(z_{0})$ is the ball taken with respect to the flat metric. Since $\int_{B_{R}(z_{0})} \mathrm{d}x \mathrm{d}y$ has growth bounded by a constant times $R$ and we are assuming $\int_{M}|K| \,\mathrm{dvol}\lt{}\infty$ we can use the dominated convergence theorem to let $R$ tend to infinity to achieve

\[ \int_{M}\left(S+\frac{1}{2} \sum_{i,j=1}^{2} h_{ij}^{2}\right) \,\mathrm{dvol}\leq \int_{M} K \,\mathrm{dvol}. \]

Recall that the Cohn–Vossen inequality states that if $M$ is a complete non-compact surface with finite total curvature, then $\int_{M} K \,\mathrm{dvol}\leq 2\pi \chi(M)$, where $\chi(M)$ is the Euler characteristic of $M$. Since $M$ is topologically a cylinder, we have $\chi(M)=0$. Thus the Cohn–Vossen inequality gives $\int_{M} K \,\mathrm{dvol}\leq 0$. Since $S+\frac{1}{2} \sum h_{ij}^{2} \geq 0$ we conclude that $M$ is totally geodesic and $S \equiv 0$ on $M$.

Hence the stability operator reduces to $\Delta-K$. By Theorem 5.1.3 there is a positive function $g$ on $M$ satisfying $\Delta g-K g=0$. Set $w=\log g$. Computing we have

\[ \Delta w=K-|\nabla w|^{2}. \]

Choosing $\zeta$ as above, we multiply by $\zeta^{2}$ and integrate by parts to get

\[ \int_{M}|\nabla w|^{2} \zeta^{2} \,\mathrm{dvol}=\int_{M} \zeta^{2} K \,\mathrm{dvol}+2 \int_{M} \zeta\langle \nabla\zeta, \nabla w\rangle \leq \int_{M} \zeta^{2} K +\frac{1}{4}|\nabla w|^{2} \zeta^{2} \,\mathrm{dvol}+4 |\nabla\zeta|^{2}. \]

The Cauchy-Schwarz inequality and the arithmetic-geometric mean inequality give

\[ 2|\zeta||\langle \nabla\zeta, \nabla w\rangle| \leq \frac{1}{4} \zeta^{2}|\nabla w|^{2}+4|\nabla\zeta|^{2}. \]

Therefore,

\[ \frac{3}{4} \int_{M}|\nabla w|^{2} \zeta^{2} \,\mathrm{dvol}\leq \int_{M} \zeta^{2} K \,\mathrm{dvol}+4 \int_{M}|\nabla\zeta|^{2} \,\mathrm{dvol}. \]

Letting $R \to \infty$ as above, we conclude that

\[ \frac{3}{4} \int_{M}|\nabla w|^{2} \,\mathrm{dvol}\leq \int_{M} K \,\mathrm{dvol}. \]

Thus $w$ is constant, so $g$ is constant and we have $K \equiv 0$.

In case $N$ has non-negative Ricci curvature, we write the stability operator as $\Delta+\mathrm{Ric}(e_{3})+\sum_{i,j=1}^{2} h_{ij}^{2}$ and note that the proof that $M$ is totally geodesic now follows as in the previous paragraph (without the assumption of finite total curvature). From the previous proof we also get that

\[ \mathrm{Ric}(e_{3})=K_{13}+K_{23}=0 \text{ on } M. \]

Since

\[ \mathrm{Ric}(e_{1})=K_{12}+K_{13} \geq 0, \quad \mathrm{Ric}(e_{2})=K_{12}+K_{23} \geq 0, \]

we have

\[ \mathrm{Ric}(e_{1})+\mathrm{Ric}(e_{2})=2 K_{12}=2 K \geq 0. \]

Thus the Gauss curvature of $M$ is non-negative and, since $M$ is a cylinder, we have $K \equiv 0$. This completes the proof of Theorem 5.1.7. ◻

The conformal Laplacian

Mon, 29 Jun 2026 00:00:00 +0000

Let $(X^n,g)$ be a closed Riemannian manifold, $n\geq3$. We use the conformal Laplacian

\[ L_g:=-c_n\Delta^g+R_g, \qquad c_n:=\frac{4(n-1)}{n-2}. \]

Here $R_g$ is the scalar curvature of $g$. With our sign convention,

\[ \int_X uL_gu\,d\mu_g = \int_X c_n|\nabla u|^2+R_gu^2\,d\mu_g. \]

\[ \widehat g=u^{\frac{4}{n-2}}g, \qquad u\gt{}0, \]

then this is the same as writing $\widehat g=e^{2f}g$ with

\[ e^{2f}=u^{\frac{4}{n-2}}, \qquad f=\frac{2}{n-2}\log u. \]

For a general conformal change $\widehat g=e^{2f}g$, the scalar curvature formula is

\begin{equation} \tag{6.1.1} \label{eq:psc-scalar-exp-transform} R_{\widehat g} = e^{-2f}\bigl( R_g-2(n-1)\Delta^gf-(n-1)(n-2)|\nabla f|^2 \bigr). \end{equation}

We now plug in $f=\frac{2}{n-2}\log u$. First,

\[ \nabla f=\frac{2}{n-2}\frac{\nabla u}{u}, \qquad |\nabla f|^2 = \frac{4}{(n-2)^2}\frac{|\nabla u|^2}{u^2}, \]

and

\[ \Delta^g f = \frac{2}{n-2}\Delta^g\log u = \frac{2}{n-2} \left( \frac{\Delta^gu}{u} - \frac{|\nabla u|^2}{u^2} \right). \]

Therefore

\[ \begin{aligned} R_{\widehat g} &= u^{-\frac{4}{n-2}} \biggl[ R_g - \frac{4(n-1)}{n-2} \left( \frac{\Delta^gu}{u} - \frac{|\nabla u|^2}{u^2} \right) - \frac{4(n-1)}{n-2} \frac{|\nabla u|^2}{u^2} \biggr] \\ &= u^{-\frac{4}{n-2}} \left( R_g-\frac{4(n-1)}{n-2}\frac{\Delta^gu}{u} \right). \end{aligned} \]

The two $|\nabla u|^2/u^2$ terms cancel exactly: the first comes from $-2(n-1)\Delta^g f$, and the second comes from $-(n-1)(n-2)|\nabla f|^2$. Since $c_n=\frac{4(n-1)}{n-2}$, we get

\begin{equation} \tag{6.1.2} \label{eq:psc-conformal-laplacian-transform} R_{\widehat g} = u^{-\frac{n+2}{n-2}} \bigl(-c_n\Delta^gu+R_gu\bigr) = u^{-\frac{n+2}{n-2}}L_gu . \end{equation}

Lemma 6.1.1

If the first eigenvalue of $L_g$ is positive, then $X$ admits a metric of positive scalar curvature in the conformal class of $g$.

Proof. Let $u\gt{}0$ be the first eigenfunction:

\[ L_gu=\lambda_1u, \qquad \lambda_1\gt{}0. \]

For $\widehat g=u^{4/(n-2)}g$, formula (6.1.2) gives

\[ R_{\widehat g} = \lambda_1u^{-\frac{4}{n-2}}\gt{}0. \]

◻

Lemma 6.1.2

Let $X^n$ be closed, $n\geq3$. If $R_g\geq0$ and $R_g\gt{}0$ somewhere, then $X$ admits a metric of positive scalar curvature.

Proof. We show that $\lambda_1(L_g)\gt{}0$. For every nonzero $\phi$,

\[ \int_X \phi L_g\phi\,d\mu_g = \int_X c_n|\nabla\phi|^2+R_g\phi^2\,d\mu_g\geq0. \]

Thus $\lambda_1\geq0$. If $\lambda_1=0$, let $u\gt{}0$ be a first eigenfunction. Then

\[ 0=\int_X c_n|\nabla u|^2+R_gu^2\,d\mu_g . \]

Both terms are nonnegative, so $\nabla u\equiv0$ and $R_gu^2\equiv0$. Since $u\gt{}0$, this forces $R_g\equiv0$, contradicting the assumption that $R_g\gt{}0$ somewhere. Hence $\lambda_1\gt{}0$, and Lemma 6.1.1 applies. ◻

Varifolds

Mon, 29 Jun 2026 00:00:00 +0000

Definition 3.1.1

An $n$-varifold $V$ in $\mathbb{R}^{n+k}$ is a Radon measure on $\mathbb{R}^{n+k}\times G(n+k,n)$, where $G(n+k,n)$ is the set of all $n$-dimensional subspaces in $\mathbb{R}^{n+k}$.

Let $M \subset \mathbb{R}^{n+k}$ be an $n$-dimensional manifold, and let $\theta$ be an $\mathcal{H}^n$-measurable function on $M$. Then the $n$-varifold $V=|(M,\theta)|$ is defined by

\[ V(U)=\int_{(x,T_xM)\in U} \theta(x) d\mathcal{H}^n|_M(x). \]

Equivalently, for any continuous compactly supported function $f$, we have

\[ V(f)=\int_{ } \theta(x)f(x,T_xM) d \mathcal{H}^n|_{M}(x). \]

Thus, a varifold arising from an $n$-dimensional manifold records tangent-plane information as well. If $\theta=1$, we usually write $|(M,1)|=|M|$. For example, $P$ is the tangent plane of $M$ at $x$ if and only if the measure $|M|$ restricted to $\{ x \} \times G(n+k,n)$ is nonzero.

Definition 3.1.2

The weight measure $\|V\|$ of an $n$-varifold $V$ is defined by

\[ \|V\|(A)=V(A \times G(n+m,n)), \]

for any Borel subset $A \subset \mathbb{R}^{n+m}$. Hence, $\|V\|$ is a Radon measure on $\mathbb{R}^{n+m}$. The support of $\|V\|$, denoted by $\mathrm{spt}\|V\|$, is defined by

\[ \mathrm{spt}\|V\|=\left\{ x \in \mathbb{R}^{n+k}: \|V\|(B^{n+k}_r(x))\gt{}0 \text{ for any }r\gt{}0 \right\}. \]

The ($n$-dimensional) density of $\|V\|$ at $x$ is defined by

\[ \Theta(\|V\|,x)=\lim_{r\to 0^+}\frac{\|V\|(B^{n+k}_r(x))}{\omega_nr^n}, \]

if the limit exists, where $\omega_n$ is the volume of the unit ball in $\mathbb{R}^n$.

Theorem 3.1.3

Suppose $V_i$ is a sequence of $n$-varifolds such that for every compact $K \subset \mathbb{R}^{n+k}$, there exists $C=C(K)$ with

\[ V_i(K \times G(n+k,n))\leq C, \]

then, up to a subsequence, we can find $V_i \to V$ in the sense of Radon measures. (The convergence is in the varifold sense.) Equivalently,

\[ \lim_{i \to \infty}V_i(f)=V(f) \]

for any $f \in C_c(\mathbb{R}^{n+k}\times G(n+k,n))$.

Suppose $V_n$ is defined by

\[ \sum_{i=1}^{2^n} |([0,1]\times \{ \frac{i}{2^n} \}, \frac{i}{2^n})|, \]

Then it converges to $V$ in the varifold sense, where

\[ V(f)=\int_{0}^{1} \int_{0}^{1} f(x,y, \{ x_2=0 \})dx dy. \]

Note that $\mathrm{spt}\|V\|=[0,1]^{2}$, so $V$ cannot be written as $V=|(M,\theta)|$ for any one-dimensional manifold $M$.

Definition 3.1.4

We say $M$ is countably $n$-rectifiable if $M \subset N \cup \bigcup_{j=1}^\infty N_j$ where $\mathcal{H}^n(N)=0$ and each $N_j$ is an $n$-dimensional embedded $C^1$ submanifold of $\mathbb{R}^{n+k}$.

Equivalently, $M$ is countably $n$-rectifiable if and only if there exists a countable family of Lipschitz maps $f_j: \mathbb{R}^n \to \mathbb{R}^{n+k}$ such that

\[ M = N \cup \bigcup_{j=1}^\infty f_j(A_j), \]

where $A_j \subset \mathbb{R}^n$ and $\mathcal{H}^n(N)=0$.

For any countably $n$-rectifiable set $M$, we write $T_xM$ for the approximate tangent space of $M$.

Definition 3.1.5

Let $M$ be an $\mathcal{H}^n$-measurable subset of $\mathbb{R}^{n+k}$ with $\mathcal{H}^n(M\cap K)\lt{}+\infty$ for every compact subset $K$. We say that an $n$-dimensional subspace $P$ is an approximate tangent space of $M$ at $x$ if and only if

\[ \lim_{r\to 0^+} \int_{ \eta_{x,r}(M)}f(y) d\mathcal{H}^n(y) = \int_{P} f(y) d\mathcal{H}^n(y),\quad \text{for any }f \in C_c(\mathbb{R}^{n+k}). \]

Theorem 3.1.6

If $M$ is rectifiable, then for $\mathcal{H}^n$-almost every $x \in M$, there exists a unique approximate tangent space $T_xM$ of $M$ at $x$.

Definition 3.1.7

We say an $n$-varifold $V$ is rectifiable if there exists a countably $n$-rectifiable, $\mathcal{H}^n$-measurable subset $M$ of $\mathbb{R}^{n+k}$ and a positive locally $\mathcal{H}^n$-integrable function $\theta$ on $M$ such that

\[ V(f)=\int_{ M} f(x,T_xM)\theta(x)d \mathcal{H}^n(x) \]

We use the notation $V=|(M,\theta)|$ for the varifold associated with $M$ and $\theta$.

Definition 3.1.8

Let $V$ be an $n$-varifold in $\mathbb{R}^{n+k}$ and let $F: \mathbb{R}^{n+k} \to \mathbb{R}^{m+l}$ be a $C^1$ map. The pushforward varifold $F_{\#}V$ is defined by

\[ F_{\#}V(\phi) = \int_{ } \phi(F(x), DF_x(S)) \left\vert J_F(x, S) \right\vert \, dV(x,S), \]

for any continuous function $\phi$ with compact support on $\mathbb{R}^{m+l} \times G(m+l, n)$, where $J_F(x, S)$ is the Jacobian of $F$ restricted to $S$, i.e.,

\[ J_F(x, S) = \sqrt{\det(\frac{\partial F}{\partial \tau_i} \cdot \frac{\partial F}{\partial \tau_j})}, \]

where $\{ \tau_i \}_{i=1}^n$ is an orthonormal basis of $S$.

Let $F_t$ be a one-parameter family of diffeomorphisms on $\mathbb{R}^{n+k}$ with $F_0$ being the identity map. The first variation of an $n$-varifold $V$ under the variation $F_t$ is defined by

\[ \left.\frac{d}{dt}\right|_{t=0} \| (F_t)_{\#} V \|(K), \]

for any compact subset $K \subset \mathbb{R}^{n+k}$.

Theorem 3.1.9

We have

\[ \left.\frac{d}{dt}\right|_{t=0} \| (F_t)_{\#} V \|(K) = \int_{ } \mathrm{div}^S \varphi(x) dV(x,S). \]

Definition 3.1.10

We define the first variation of $V$ as a linear functional $\delta V$ on $C_c(\mathbb{R}^{n+k}, \mathbb{R}^{n+k})$ by

\[ \delta V(\varphi) = \int_{ } \mathrm{div}^S \varphi(x) dV(x,S). \]

Definition 3.1.11

We say that $V$ has bounded first variation if $\delta V$ is a bounded linear functional on $C_c(\mathbb{R}^{n+k}, \mathbb{R}^{n+k})$.

\[ \delta V(\varphi) = \int \langle \nu_V(x), \varphi(x) \rangle d\|\delta V\|(x). \]

In particular, we can decompose $\|\delta V\|=h\|V\|+\sigma_V$ into the absolutely continuous part $h\|V\|$ and the singular part $\sigma_V$ with respect to $\|V\|$.

Let $\mu$ and $\nu$ be two Radon measures on $\mathbb{R}^{n+k}$. We say $\mu$ is absolutely continuous with respect to $\nu$ (denoted $\mu \ll \nu$) if for every Borel set $A$, $\nu(A) = 0$ implies $\mu(A) = 0$. By the Radon-Nikodym theorem, if $\mu \ll \nu$, then there exists a $\nu$-measurable function $h$ such that

\[ \mu(A) = \int_A h \, d\nu \]

for all Borel sets $A$.

Conversely, a measure $\mu$ is singular with respect to $\nu$ if there exists a Borel set $A$ such that $\nu(A) = 0$ and $\mu(\mathbb{R}^{n+k} \setminus A) = 0$. Intuitively, $\mu$ and $\nu$ are supported on disjoint sets.

Any Radon measure $\mu$ can be uniquely decomposed as

\[ \mu = \mu_{\mathrm{ac}} + \mu_{\mathrm{sing}}, \]

where $\mu_{\mathrm{ac}} \ll \nu$ and $\mu_{\mathrm{sing}} \perp \nu$. So, if $V$ has bounded first variation, we can write

\[ \delta V(\varphi) =-\int \langle \vec{H}, \varphi(x) \rangle d\|V\|(x) + \int \langle \nu_V(x), \varphi(x) \rangle d\sigma_V(x). \]

where $\vec{H} =-h \nu_V$ is called the generalized mean curvature vector of $V$, and $\nu_V\sigma_V$ is called the generalized boundary of $V$.

Suppose $M=\{ (\cos \theta, \sin \theta): \theta \in (0,\pi) \}$, the 1-dimensional half-circle in $\mathbb{R}^2$, and $V=|M|$ is the associated varifold. Then, we have

\[ \begin{aligned} \delta V(\varphi)={}&\int_{ M} \mathrm{div}^M \varphi(x) d\mathcal{H}^1|_M(x)=\int_{ M} \mathrm{div}^M \varphi^\top(x) d\mathcal{H}^1|_M(x)+\int_{ M} \varphi(x) \cdot x\\ ={}& \varphi(1,0) \cdot (0,-1) + \varphi(-1,0) \cdot (0,-1) + \int_{ M} \varphi(x) \cdot x d\mathcal{H}^1|_M(x). \end{aligned} \]

So the generalized mean curvature vector $\vec{H}=-x$ is just the usual mean curvature vector of $M$, and the generalized boundary $\nu_V\sigma_V=(0,-1)\delta_{(1,0)} + (0,-1)\delta_{(-1,0)}$ represents the two boundary points of $M$ with the corresponding outward normal vectors.

Suppose $M=[0,1]\times \{ 0 \}, \theta(x,y)=x$, and $V=|(M,\theta)|$ is the associated varifold. Then, we have

\[ \delta V = \int_{ 0}^1 \frac{\partial }{\partial x} \varphi^\top(x,0) x dx = \varphi^\top(1,0)\cdot (1,0) - \int_{ 0}^1 \varphi^\top(x,0)\cdot (1,0) dx. \]

So we have

\[ \vec{H}=(\frac{1}{x},0),\quad \nu_V\sigma_V=(1,0)\delta_{(1,0)}. \]

Thus, the generalized mean curvature vector $\vec{H}$ depends not only on the geometry of $M$, but also on the weight function $\theta$.

This example also shows that, unlike the case of smooth submanifolds, the generalized mean curvature vector of a varifold may not be perpendicular to the tangent plane. If we restrict to integral rectifiable varifolds, we have the following result of Brakke [Bra15].

Theorem 3.1.12

Suppose $V$ is an integral rectifiable varifold with bounded first variation, then the generalized mean curvature vector $\vec{H}$ is perpendicular to the tangent plane of $V$ for $\|V\|$-almost every point.

Definition 3.1.13

We say that $V$ is a stationary varifold in $U$ if for any $\varphi \in C_c(U,\mathbb{R}^{n+k})$, we have

\[ \int_{ } \mathrm{div}^S \varphi(x)dV(x,S)=0. \]

If $V=|M|$, then $V$ being stationary means

\[ \int_{ M} \mathrm{div}^M \varphi(x) d\mathcal{H}^n|_M(x)=0. \]

This is equivalent to saying that $M$ is minimal.

This notion of stationarity is very weak. Triple-junctions of three half-planes meeting at $120$ degrees are stationary.

Theorem 3.1.14

Let $V$ be a stationary $n$-varifold in $\mathbb{R}^{n+k}$. Then the function

\[ \Phi(r) = \frac{\|V\|(B_r(x_0))}{\omega_n r^n} \]

is monotone increasing in $r \gt{} 0$. Moreover, we have

\[ \Phi(r)-\Phi(s)=\int_{ B_r(x_0)\backslash B_s(x_0)} \frac{|(y-x_0)^\perp|^2}{|y-x_0|^{n+2}} d\|V\|(y) \]

Proof. We can choose test vector fields $\varphi$ as before. Suppose $x_0=0$ for simplicity.

\[ \varphi = \begin{cases} y\left( \frac{1}{|y|^n}-\frac{1}{\rho^n} \right), & \sigma\leq |y|\leq \rho\\ y\left( \frac{1}{\sigma^n}-\frac{1}{\rho^n} \right), & |y|\lt{}\sigma\\ \end{cases} \]

We insert $\varphi$ into the first variation formula and get the desired monotonicity formula. ◻

One can obtain a modified monotonicity formula for varifolds with $L^p$-integrable mean curvature ($p\gt{}n$) and no generalized boundary.

Theorem 3.1.15

Suppose $V_i$ is a sequence of rectifiable stationary varifolds in $U$ and for any $K \subset \subset U$, $\sup\|V_i\|(K)\lt{}+\infty$. We also assume $\Theta(\|V_i\|,x)\geq 1$ for almost every $x\in \mathrm{spt}\|V_i\|$. Then, up to a subsequence, there exists a rectifiable stationary varifold $V$ in $U$ such that $V_i\to V$ in the varifold sense.

In particular, if each $V_i$ is integral, so is $V$.

The condition $\Theta\geq 1$ is essential.
Stationarity is also essential: $V_n=\sum_{i=1}^{2^n}|[\frac{2i-1}{2^{n+1}}, \frac{2i}{2^{n+1}}]|$ converges to $|([0,1],\frac{1}{2})|$.

Definition 3.1.16

The tangent cone of an $n$-varifold $V$ at $x$, denoted by $\mathrm{VarTan}(V,x)$, is defined by

\[ \mathrm{VarTan}(V,x):= \{V':V'=\lim_{i\to \infty}(\eta_{x,\rho_i})_\#V\text{ for some }\rho_i\to 0^+ \}. \]

The tangent cone at infinity, denoted by $\operatorname{VarTan}(V,\infty)$, is defined by

\[ \mathrm{VarTan}(V,\infty):= \{V':V'=\lim_{i\to \infty}(\eta_{0,\rho_i})_\#V\text{ for some }\rho_i\to +\infty\}. \]

The tangent cone of a varifold may not be unique. Right now, we do not even know if the tangent cone of any stationary rectifiable varifold is unique or not. This is still an open problem.

Proposition 3.1.17

If $V$ is stationary in $\mathbb{R}^{n+k}$, then the tangent cone of $V$ at $\infty$ or $x$ is a stationary cone.

Finally, for any $n$-varifold defined on $U$, $\mathrm{reg}\|V\|$ denotes the regular set of $\mathrm{spt}\|V\|$ in $U$, i.e., the set of points $x \in \mathrm{spt}\|V\|$ such that there exists $r\gt{}0$ with $B^{n+1}_r(x)\subset U$ and $\mathrm{spt}\|V\|\cap B^{n+1}_r(x)$ is a smooth (immersed) hypersurface in $B^{n+1}_r(x)$. $\mathrm{sing}\|V\|$ denotes the singular set of $\mathrm{spt}\|V\|$ in $U$.

Allard’s regularity theorem is a foundational result in geometric measure theory. It gives conditions under which a stationary varifold is regular (i.e., smooth) near a point.

Theorem 3.1.18

Let $V$ be an $n$-dimensional stationary integral rectifiable varifold in $B_2(0)\subset \mathbb{R}^{n+k}$, and suppose that

\[ \|V\|(B_2(0))\leq (1+\delta)\omega_n 2^n \]

for some $0\lt{}\delta\lt{}1$. Then, there exists $\varepsilon=\varepsilon(n,\delta)$ such that if

\[ E^{2}:=\int_{ B_2(0)} \operatorname{dist}^2(x,P)\,d\|V\|(x)\leq \varepsilon \]

where $P=\{ x_{n+1}=0 \}$, then there exists a function $u \in C^{1,\alpha}(\bar{B_1^n(0)})$ such that

\[ \mathrm{spt}\|V\| \cap B_1(0) = \{ (x',u(x')): x'\in B_1^n(0) \} \]

and

\[ \sup_{B_1^n(0)} |u| + \|D u\|_{L^{\infty}(B_1^n(0))} + [D u]_{C^{0,\alpha}(B_1^n(0))} \leq C(n,\delta) E. \]

Note that $\delta$ cannot be 1 since otherwise, we have the scaled catenoid as a counterexample.

Theorem 3.1.19

Let $V$ be an $n$-dimensional stationary integral rectifiable varifold in an open set $U \subset \mathbb{R}^{n+k}$. Suppose that the density of $\|V\|$ at a point $x_0 \in U$ is $1$. Then $x_0\in \mathrm{reg}\|V\|$.

2. The capillary prototype

Thu, 27 Aug 2026 00:00:00 +0000

The point of departure was not a ready-made formula, but a constrained search problem in our stable-capillary work. Let $M^n$ be a capillary minimal hypersurface, let $\nu$ be its unit normal, and write

\[ p_\pm=\cos\theta\,e_1\pm\sin\theta\,e_2. \]

Here $e_2$ is the unit vector orthogonal to $e_1$ in the two-plane containing the prescribed normal directions, and we write

\[ \nu_1=\ip{\nu}{e_1}, \qquad \nu_2=\ip{\nu}{e_2}. \]

We wanted one nonnegative scalar function $g=g(\nu)$ satisfying all three of the following requirements:

a strictly positive Schoen-type inequality

\[ \tag{2.1}\label{eq:capillary-search-inequality} g\Delta_M g+\abs{A}^2g^2\geq c\abs{A}^2 \qquad\text{on }\{g\gt{}0\}, \]

for some $c\gt{}0$;

the exact zero set

\[ g(\nu)=0\quad\Longleftrightarrow\quad\nu\in\{p_+,p_-\}; \]

the Robin boundary condition

\[ \tag{2.2}\label{eq:capillary-robin} B_\theta(U) \coloneqq\partial_\eta U +2\cot\theta\,A(\eta,\eta)U, \qquad B_\theta(g^2)=0 \qquad\text{on }\partial M\cap\{g\gt{}0\}. \]

Here and below, $\eta$ denotes the outer unit conormal of $\partial M$ in $M$.

2.1 The boundary condition as a design equation

The Robin condition was not an afterthought. It was one of the main filters on every candidate. Along $\partial M$,

\[ \tag{2.3}\label{eq:capillary-boundary-identities} \nu_1=\cos\theta, \qquad \partial_\eta\nu_1 =\sin\theta\,A(\eta,\eta), \qquad \partial_\eta\nu_2 =-\cot\theta\,A(\eta,\eta)\nu_2. \]

Consequently, if a one-variable squared profile $U=p^2(\nu_1)$ is positive on the boundary, then direct substitution into (2.2) shows that its square-root profile satisfies

\[ \tag{2.4}\label{eq:log-p-boundary-condition} (\log p)'(\cos\theta) =-\frac{\cos\theta}{\sin^2\theta}. \]

This logarithmic-derivative constraint, evaluated at $\nu_1=\cos\theta$, was used to generate the one-variable candidates.

There was also an elementary but decisive closure property. The operator $B_\theta$ is linear, and hence

\[ B_\theta(U_1)=B_\theta(U_2)=0 \quad\Longrightarrow\quad B_\theta(\alpha_1U_1+\alpha_2U_2)=0 \qquad\text{for all }\alpha_1,\alpha_2\in\R. \]

Thus squared profiles could be designed additively without introducing a second boundary operator.

2.2 What was tried before the final family

Before reaching the final capillary family, we tried many alternatives. We began with the natural two-pole detector

\[ g_{\mathrm{pair}} =\left[(1-\ip{\nu}{p_+})(1-\ip{\nu}{p_-})\right]^{1/2}. \]

It has exactly the desired two zeros and satisfies $B_\theta(g_{\mathrm{pair}}^2)=0$. On the other hand, the one-pole background

\[ h_\theta=1-\cos\theta\,\nu_1 \]

satisfies the exceptionally clean identities

\[ (\Delta_M+\abs{A}^2)h_\theta=\abs{A}^2, \qquad B_\theta(h_\theta^2)=0, \]

but it does not detect the two prescribed normals. Thus one candidate had the right zero set, while the other had the cleaner differential inequality.

We next tried to interpolate between these advantages through the geometric mixture

\[ g=g_{\mathrm{pair}}^\alpha h_\theta^{1-\alpha}, \qquad 0\lt{}\alpha\lt{}1, \]

as well as truncated functions, power profiles, and profiles $p(\nu_1)=\exp(q(\nu_1))$ chosen to satisfy (2.4). Working initially in the acute-angle range $0\lt{}\theta\lt{}\pi/2$, we tested, among others:

\[ q(t)=-\frac{\cos\theta}{\sin^2\theta}t, \qquad q(t)=-\frac{t^a}{a\cos^{a-1}\theta\sin^2\theta}, \qquad q(t)=\log\left(1-\frac{\cos^{2-b}\theta} {b\sin^2\theta+\cos^2\theta}t^b\right). \]

We also tested the direct power $1-\cos^{2-a}\theta\,\nu_1^a$. The calculations explained why these formulas were not the final choice: the multiplicative mixtures created an unfavorable cross-gradient term, while the one-variable modifiers could not simultaneously provide the exact two-pole zero set and a uniform interior margin.

At this stage, hand calculations were repeatedly converted into symbolic and numerical tests. For a chosen profile and parameter, we varied the normal direction and the trace-free second fundamental form, then computed the smallest value of the normalized expression in (2.1). The experiments were especially useful for rejecting a profile when its minimum became negative and for identifying a small-parameter regime that was worth proving analytically.

2.3 The additive route to the final family

The addition rule for $B_\theta$ led directly to the final family. In addition to $g_{\mathrm{pair}}$, define $D(\nu)=1-\nu_1^2$. The three building blocks are $D(\nu)$, $g_{\mathrm{pair}}^2(\nu)$, and $\nu_2^2$. Using (2.3), a direct computation gives $B_\theta(D)=B_\theta(g_{\mathrm{pair}}^2)=B_\theta(\nu_2^2)=0$. Consequently, every function of the form

\[ U_\alpha(\nu) =\alpha_1D(\nu)+\alpha_2g_{\mathrm{pair}}^2(\nu) +\alpha_3\nu_2^2 \]

satisfies $B_\theta(U_\alpha)=0$.

At either prescribed normal $p_\pm$, we have $D(p_\pm)=\nu_2^2(p_\pm)=\sin^2\theta$ and $g_{\mathrm{pair}}(p_\pm)=0$. Thus $U_\alpha(p_\pm)=0$ precisely when $\alpha_1+\alpha_3=0$. Taking $\alpha_3=-\alpha_1$, and using

\[ g_{\mathrm{pair}}^2 =\sin^2\theta(1-\nu_1^2-\nu_2^2) +(\cos\theta-\nu_1)^2, \]

we obtain

\[ U_\alpha(\nu) =(\alpha_1+\alpha_2\sin^2\theta) (1-\nu_1^2-\nu_2^2) +\alpha_2(\cos\theta-\nu_1)^2. \]

If $\alpha_2\gt{}0$ and $C_\alpha=\alpha_1+\alpha_2\sin^2\theta\gt{}0$, this expression is nonnegative on the unit sphere and vanishes exactly when $\nu_1=\cos\theta$ and $\nu_2=\pm\sin\theta$, that is, precisely at $p_+$ and $p_-$. Dividing by the harmless positive constant $C_\alpha$, the family can be written using the more transparent parameter $k=\frac{\alpha_2}{C_\alpha}\gt{}0$ as

\[ \class{key-formula-math}{g_{\theta,k}^2 =1-\nu_1^2-\nu_2^2+k(\cos\theta-\nu_1)^2.} \]

Its first part confines the normal to the two-plane containing $p_\pm$, but cannot distinguish points inside that plane. The second part selects the two prescribed normals, at the cost of introducing the difficult mixed terms. The parameter $k$ controls their size: the margin left by the first part absorbs them when $k$ is in the appropriate range.

We then tested this family numerically with very small values of $k$. The experiments suggested that a positive lower bound should indeed exist, although the observed margin decreased more rapidly as $\theta$ became small. This numerical evidence identified a plausible regime, but did not prove uniform positivity. The subsequent rigorous calculation verified analytically that the required positive lower bound exists.

2.4 The design blueprint from the capillary problem

The capillary construction suggested more than one successful formula. It provided a general architecture for the next stage of the search. At this point we regarded

\[ \tag{2.5}\label{eq:first-general-ansatz} S+kP \]

only as a prototype, with the following division of roles.

The background $S$. This term should force the normal into a preferred low-dimensional set, usually a low-dimensional span. It should also provide the robust part of the differential inequality. By itself, however, it need not distinguish the prescribed normal directions inside that set.
The detector $P$. This term should separate the individual prescribed directions left indistinguishable by $S$, and ultimately produce the desired finite zero set. Its derivatives are typically much harder to control than those of the background.
The small parameter $k$. The coefficient of $P$ should be chosen small enough that the complicated terms introduced by the detector can be absorbed by the margin supplied by $S$. The parameter is therefore part of the design, rather than a final cosmetic adjustment.
Freedom in the outer form. The general problem no longer carries the capillary Robin boundary condition. We were therefore not restricted to taking the test function to be exactly a square root of $S+kP$. Powers such as $(S+kP)^\gamma$, product forms, normalized products, and related compositions were all legitimate candidates, provided that they retained the correct zero set and differential positivity.

This became the blueprint for the subsequent experiments: first choose a simple $S$ that enforces low-dimensionality, then vary the detector $P$, the parameter $k$, and the outer power or product structure. Symbolic and numerical tests could then identify which versions preserved a positive margin and, equally importantly, where each proposed design failed.

Classification of Stable Minimal Cones for n<= 6

Mon, 29 Jun 2026 00:00:00 +0000

Proposition 2.2.1

Suppose $\mathbf C$ is a minimal cone in $\mathbb{R}^{n+1}$, smooth away from the origin. Then we have the following Simons-type inequality on $\mathbf C\setminus\{0\}$:

\[ \frac{1}{2}\Delta |A|^{2} \geq -|A|^4 + 2 \frac{|A|^{2}}{r^{2}}+|\nabla |A||^{2}. \]

Proof. Choose $\{ e_i \}$ to be a local orthonormal frame on $\mathbf C\setminus\{0\}$ such that $e_n$ is the radial direction $\partial_r$. Recall the Simons identity for minimal hypersurfaces in $\mathbb{R}^{n+1}$:

\[ \frac{1}{2}\Delta |A|^{2} = -|A|^4 + |\nabla A|^2. \]

It suffices to show

\[ |\nabla A|^2 \geq 2 \frac{|A|^2}{r^2}+|\nabla |A||^2. \]

Since $\mathbf C$ is a cone, we have $A_{in}=0$ and $A_{ij,n}=-\frac{1}{r}A_{ij}$ for $i,j\lt{}n$. Indeed,

\[ A(rp)=\frac{A(p)}{r}. \]

Taking derivative in the radial direction gives

\[ A_{ij,n} (rp)= - \frac{A_{ij}(p)}{r^{2}}= -\frac{1}{r}A_{ij}(rp). \]

So we have

\[ \sum_{i,j,k}^{}A_{ij,k}^{2}=\sum_{\alpha,\beta=1}^{n-1}3 A_{\alpha\beta,n}^{2} + \sum_{\alpha,\beta,\gamma=1}^{n-1}A_{\alpha\beta,\gamma}^{2} = 2 \frac{|A|^2}{r^2} + \sum_{\alpha,\beta=1}^{n-1}A_{\alpha\beta,n}^{2}+ \sum_{\alpha,\beta,\gamma=1}^{n-1}A_{\alpha\beta,\gamma}^{2}. \]

At a fixed point, choose $\{e_\alpha\}_{\alpha=1}^{n-1}$ so that

\[ A_{\alpha\beta}=\lambda_\alpha\delta_{\alpha\beta}. \]

Then

\[ |\nabla |A||^2 =\frac{1}{|A|^2}\sum_{k=1}^{n}\Big(\sum_{\alpha=1}^{n-1}\lambda_\alpha A_{\alpha\alpha,k}\Big)^2 =\frac{1}{|A|^2}\sum_{\beta=1}^{n-1}\Big(\sum_{\alpha=1}^{n-1}\lambda_\alpha A_{\alpha\alpha,\beta}\Big)^2 +\frac{1}{|A|^2}\Big(\sum_{\alpha=1}^{n-1}\lambda_\alpha A_{\alpha\alpha,n}\Big)^2. \]

By Cauchy–Schwarz and $A_{\alpha\alpha,n}=-\frac1rA_{\alpha\alpha}$,

\[ |\nabla |A||^2 \leq \sum_{\alpha,\beta=1}^{n-1}A_{\alpha\alpha,\beta}^2+\frac{|A|^2}{r^2} \leq \sum_{\alpha,\beta,\gamma=1}^{n-1}A_{\alpha\beta,\gamma}^2+\sum_{\alpha,\beta=1}^{n-1}A_{\alpha\beta,n}^2. \]

Therefore,

\[ |\nabla A|^2 =2\frac{|A|^2}{r^2} +\sum_{\alpha,\beta=1}^{n-1}A_{\alpha\beta,n}^2 +\sum_{\alpha,\beta,\gamma=1}^{n-1}A_{\alpha\beta,\gamma}^2 \geq 2\frac{|A|^2}{r^2}+|\nabla |A||^2. \]

This completes the proof. ◻

A similar computation yields

\[ \frac{1}{2}\Delta|A|^{2}\geq p |\nabla |A||^{2} + (3-p)\frac{|A|^{2}}{r^{2}} - |A|^4, \]

for any $p\leq 1+\frac{2}{n-1}$.

Theorem 2.2.2

Let $\mathbf C^n \subset \mathbb{R}^{n+1}$ be a minimal nonflat cone, smooth away from the origin. If $n \leq 6$, $\mathbf C$ is unstable. If $n \geq 7$, there exists a stable minimal nonflat cone.

Proof. Suppose $\mathbf C$ is stable. Testing the stability inequality with $|A|\varphi$, we get

\[ \begin{aligned} \int_{ } |A|^{4}\varphi^{2} \leq{}& \int_{ } |\nabla (|A|\varphi)|^2 = \int_{ } |\nabla |A||^2 \varphi^2 + |A|^2 |\nabla \varphi|^2 + \frac{1}{2} \langle\nabla |A|^{2}, \nabla \varphi^{2}\rangle\\ \leq{}& \int_{ } |\nabla |A||^2 \varphi^2 + |A|^{2}|\nabla \varphi|^2 - \frac{1}{2}\varphi^{2}\Delta |A|^{2} \\ \leq{}& \int_{ } |A|^{2}|\nabla \varphi|^2 + \varphi^{2}|A|^4 - 2 \frac{|A|^{2}}{r^{2}}\varphi^{2}. \end{aligned} \]

Hence, we obtain

\[ 2 \int_{ } \frac{|A|^{2}}{r^{2}}\varphi^{2} \leq \int_{ } |A|^{2}|\nabla \varphi|^2. \]

Now choose

\[ \varphi= \max\{ 1,r \}^{1-\frac{n}{2}-2\varepsilon} r^{1+\varepsilon}. \]

We need to verify that this function is admissible. We compute

\[ \begin{aligned} \int_{ } |A|^{2}|\nabla \varphi|^2={}&(1+\varepsilon)^{2}\int_{ \{ r\lt{}1 \}}|A|^{2} r^{2\varepsilon}+(2-\frac{n}{2}-\varepsilon)^{2}\int_{ \{ r\geq 1 \}} |A|^{2} r^{-n+2-\varepsilon}\\ ={}&(1+\varepsilon)^{2}\int_{ 0}^1dr \int_{ \Sigma} |A_\Sigma|^{2} r^{2\varepsilon-2+n-1}d\Sigma\\ &+(2-\frac{n}{2}-\varepsilon)^{2}\int_{ 1}^\infty dr \int_{ \Sigma} |A_\Sigma|^{2} r^{-n-2\varepsilon+n-1}d\Sigma\\ ={}&(\int_{ 0}^1 r^{2\varepsilon+n-3}dr + (2-\frac{n}{2}-\varepsilon)^{2}\int_{ 1}^\infty r^{-2\varepsilon-1}dr)\int_{ \Sigma} |A_\Sigma|^{2} d\Sigma\lt{}+\infty \end{aligned} \]

Here, $\Sigma=\mathbf C\cap S^n$ is the link of the cone, which is a smooth closed minimal hypersurface in $S^n$. Hence, $\varphi$ is admissible. On the other hand, we have

\[ \begin{aligned} 2\int_{ } \frac{|A|^{2}}{r^{2}}\varphi^{2}={}& 2\int_{ \{ r\lt{}1 \}} |A|^{2} r^{2\varepsilon} + 2\int_{ \{ r\geq 1 \}} |A|^{2} r^{-n+2-4\varepsilon}\\ \end{aligned} \]

When $n\leq 6$, we can choose $\varepsilon\gt{}0$ small enough such that $2\gt{}(1+\varepsilon)^{2}$ and $2\gt{}(2-\frac{n}{2}-\varepsilon)^{2}$. Hence $|A|^2\equiv 0$, so $\mathbf C$ is flat, a contradiction. Therefore, $\mathbf C$ is unstable.

It remains to construct a stable minimal nonflat cone in $\mathbb{R}^8$.

Define the Simons-type cone by

\[ \mathbf C_{p,q} =\left\{(x,y)\in \mathbb{R}^{p+1}\times \mathbb{R}^{q+1}:\ q|x|^2=p|y|^2\right\}. \]

Lemma 2.2.3

$\mathbf C_{p,q}$ is a minimal cone in $\mathbb{R}^{p+q+2}$, smooth away from the origin, and it is stable if and only if $p+q\geq 6$.

One can verify directly that $\mathbf C_{p,q}$ is minimal by computing its mean curvature. In particular, its principal curvatures are

\[ \kappa_1=\cdots=\kappa_p=\sqrt{\frac{q}{p}}, \qquad \kappa_{p+1}=\cdots=\kappa_{p+q}=-\sqrt{\frac{p}{q}},\quad \text{and}\quad \kappa_{p+q+1}=0. \]

Its second fundamental form satisfies $|A|^{2}=\frac{n-1}{r^{2}}$. Choose $X= \frac{\varphi^{2}}{r^{2}}x$ and insert it into the first variation formula. Then

\[ 0=\int_{ } \mathrm{div}^{\mathbf C_{p,q}}(X)=\int_{ } n\frac{\varphi^{2}}{r^{2}}-2\frac{\varphi^{2}}{r^{4}}|x|^2+2 \frac{\varphi \nabla^{\mathbf C_{p,q}} \varphi \cdot x}{r^{2}}=\int_{ } (n-2)\frac{\varphi^{2}}{r^{2}}+2 \frac{\varphi \nabla^{\mathbf C_{p,q}} \varphi \cdot x}{r^{2}}. \]

Hence

\[ \int_{ } (n-2) \frac{\varphi^{2}}{r^{2}}\leq 2 \sqrt{\int_{ } \frac{\varphi^{2}}{r^{2}} \int_{ } |\nabla^{\mathbf C_{p,q}} \varphi|^2} \implies \int_{ } \frac{(n-2)^{2}}{4} \frac{\varphi^{2}}{r^{2}} \leq \int_{ } |\nabla^{\mathbf C_{p,q}} \varphi|^2. \]

Since $|A|^2=\frac{n-1}{r^2}$, this can be rewritten as

\[ \int_{ } \frac{(n-2)^{2}}{4(n-1)}|A|^{2} \varphi^{2} \leq \int_{ } |\nabla^{\mathbf C_{p,q}} \varphi|^2. \]

In particular, if $n\geq 7$, then $\frac{(n-2)^2}{4(n-1)}\geq 1$, so $\mathbf C_{p,q}$ is stable. ◻

In fact, one has the stronger result:

Theorem 2.2.4

Each $\mathbf C_{p,q}$ is area-minimizing if $p+q\geq 6$ except for the case $p,q= (1,5)$ or $(5,1)$.

The case $\mathbf C_{p,p}$ with $p\geq 3$ was proved by Bombieri–De Giorgi–Giusti [BDGG69]. Lawson [Law72] later proved the result for $p+q\gt{}6$, and Simões [Sim74] handled the remaining cases $p,q=(2,4)$ or $(4,2)$.

The original proof is quite involved and relies on calibrations. One seeks a vector field $\xi$ such that $\operatorname{div}^{\mathbf C} \xi=0$ and $\xi=\nu_{\mathbf C}$ on $\mathbf C\setminus\{0\}$, which implies that $\mathbf C$ is area-minimizing. Finding such a calibration for $\mathbf C_{p,q}$ is difficult and requires solving an ODE.

Here we present a different proof due to De Philippis–Paolini [DPP09], which uses a sub-calibration argument together with an explicit construction of a sub-calibration for $\mathbf C_{p,p}$.

Conformal descent on a stable minimal hypersurface

Mon, 29 Jun 2026 00:00:00 +0000

Let $(N^{n+1},g_N)$ be a closed Riemannian manifold and let $\Sigma^n\subset N$ be a closed, two-sided, stable minimal hypersurface. Denote the induced metric on $\Sigma$ by $g$. The stability inequality is

\begin{equation} \tag{6.2.1} \label{eq:psc-stability-hypersurface} \int_\Sigma |\nabla\phi|^2\,d\mu_g \geq \int_\Sigma \bigl(|A|^2+\mathrm{Ric}_N(\nu,\nu)\bigr)\phi^2\,d\mu_g, \qquad \phi\in C^\infty(\Sigma). \end{equation}

The Gauss equation, using $H=0$, gives

\begin{equation} \tag{6.2.2} \label{eq:psc-gauss-scalar} R_N = R_\Sigma+2\mathrm{Ric}_N(\nu,\nu)+|A|^2. \end{equation}

Combining (6.2.1) and (6.2.2),

\begin{equation} \tag{6.2.3} \label{eq:psc-sy-rearrangement} \int_\Sigma 2|\nabla\phi|^2+R_\Sigma\phi^2\,d\mu_g \geq \int_\Sigma (R_N+|A|^2)\phi^2\,d\mu_g . \end{equation}

This is the basic Schoen–Yau rearrangement.

Proposition 6.2.1

Assume $R_N\gt{}0$ and $n\geq3$. Then every closed, two-sided, stable minimal hypersurface $\Sigma^n\subset N^{n+1}$ admits a metric of positive scalar curvature.

Proof. Since $\Sigma$ is compact and $R_N\gt{}0$ on $N$, there is a number $\kappa\gt{}0$ such that $R_N\geq\kappa$ on $\Sigma$. From (6.2.3),

\[ \int_\Sigma 2|\nabla\phi|^2+R_\Sigma\phi^2\,d\mu_g \geq \kappa\int_\Sigma \phi^2\,d\mu_g . \]

Because $c_n=4(n-1)/(n-2)\geq2$, we also have

\[ \int_\Sigma c_n|\nabla\phi|^2+R_\Sigma\phi^2\,d\mu_g \geq \kappa\int_\Sigma \phi^2\,d\mu_g . \]

Thus the first eigenvalue of the conformal Laplacian $L_g$ is positive. The claim follows from Lemma 6.1.1. ◻

When $n=2$, the same rearrangement gives useful topological information directly. Taking $\phi\equiv1$ in (6.2.3) gives

\[ \int_\Sigma R_\Sigma\,d\mu_g = 2\int_\Sigma K_\Sigma\,d\mu_g \gt{}0. \]

Hence $\chi(\Sigma)\gt{}0$ by Gauss–Bonnet. In particular a closed orientable stable minimal surface in a three-manifold with $R_N\gt{}0$ is a union of two-spheres.

Higher-Dimensional Stable Bernstein Theorems

Mon, 29 Jun 2026 00:00:00 +0000

We outline the strategy for the unconditional stable Bernstein theorem in dimensions $n=3,4,5$. The argument has three main steps: a conformal change producing positive spectral bi-Ricci curvature, the construction of $\mu$-bubbles, and uniform area bounds for these bubbles via spectral Ricci estimates. The conformal and spectral curvature estimates are proved in §5.2.1; the $\mu$-bubble reduction is treated in §5.2.2, and the spectral Ricci estimates are recorded in §5.2.6.

Step 1: Conformal change and spectral bi-Ricci curvature. Let $M^n\hookrightarrow \mathbb{R}^{n+1}$ be a complete, two-sided, stable minimal immersion. Fix $0\in M$ and write $r(x)=|x|$ for the Euclidean distance from the origin. Following Schoen–Simon–Yau and the recent stable Bernstein literature, we pass to the conformal metric

\[ \tilde{g}=r^{-2}g, \qquad g=\varphi^*\delta, \]

where $g$ is the induced metric. Stability of $M$ yields a spectral inequality for the conformal Laplacian with a curvature potential. To state it uniformly in dimension, we introduce the bi-Ricci curvature.

Definition 5.2.1

Let $(N,g)$ be a Riemannian manifold of dimension $m\geq 2$. For orthonormal $v,w\in T_pN$, define

\[ \mathrm{BiRic}(v,w):=\mathrm{Ric}(v,v)+\mathrm{Ric}(w,w)-R(v,w,v,w). \]

As functions on $N$, we write $\mathrm{BiRic}$ and $\mathrm{Ric}$ for the minimum over unit directions (with $v\perp w$ for $\mathrm{BiRic}$):

\[ \mathrm{BiRic}(p):=\inf_{\substack{v,w\in T_pN\\ |v|=|w|=1,\ \langle v,w\rangle=0}} \mathrm{BiRic}(v,w), \qquad \mathrm{Ric}(p):=\inf_{\substack{v\in T_pN\\ |v|=1}} \mathrm{Ric}(v,v). \]

In dimension $m=3$, the bi-Ricci curvature is independent of the chosen orthonormal pair:

\[ \mathrm{BiRic}(v,w)=\frac12 R. \]

This is why the three-dimensional part of the argument is often stated as a spectral scalar-curvature condition.

We use spectral curvature condition to mean a lower bound for the first eigenvalue of a Schrödinger operator whose potential is built from one of these curvature quantities. The coefficient depends on the dimension and on the normalization used in the $\mu$-bubble argument.

In the conformal metric $\tilde g=r^{-2}g$, stability gives the concrete inputs needed below: for $n=3$,

\[ \lambda_1\left(-\tilde{\Delta}+\frac12\tilde R\right)\geq \frac32 \]

by the conformal scalar-curvature computation in §5.2.1, while for $n=4$,

\[ \lambda_1\left(-\tilde{\Delta}-\widetilde{\mathrm{BiRic}}\right)\geq 1 \]

after the corresponding conformal bi-Ricci normalization.

Thus, after conformal reparametrization, the problem is reduced to studying a manifold with non-negative spectral bi-Ricci curvature (in dimension three this is precisely non-negative spectral scalar curvature). This replaces the extrinsic curvature input of the minimal hypersurface by an intrinsic spectral curvature hypothesis on $(M,\tilde{g})$.

Step 2: Construction of $\mu$-bubbles. The second step is to produce separating hypersurfaces inside a collar of $(M,\tilde{g})$ with controlled geometry. These are $\mu$-bubbles.

Classically, a soap bubble is a hypersurface of constant mean curvature (CMC) enclosing a prescribed volume—the archetypal prescribed mean curvature problem. More generally, one studies hypersurfaces whose mean curvature is a prescribed function of position and geometry; Gromov introduced and systematically used such constructions in his work on scalar curvature, naming them $\mu$-bubbles (the name reflects the measure $\mu$ specifying the prescription). In the present setting, one considers a Riemannian manifold $(N,g)$ with boundary $\partial N=\partial_+N\cup\partial_-N$ and seeks a domain $\Omega\subset N$ whose boundary $\Sigma=\partial\Omega\setminus\partial_-N$ (the $\mu$-bubble) solves a variational problem of prescribed mean curvature type: schematically, one minimizes a functional of the form

\[ \mathcal{A}(\Omega)=\int_{\Sigma} w\,d\mathcal{H}^{m-1}-\int_{\Omega} w\,h\,d\mathcal{H}^m \]

for a weight $w$ (often the first eigenfunction of a suitable operator on $N$) and a carefully chosen function $h$ determined by the curvature of $N$. Under the spectral bi-Ricci condition from Step 1, such a minimizer exists and provides a $\mu$-bubble $\Sigma$ separating $\partial_-N$ from $\partial_+N$ (Theorem 5.2.8).

In recent years, $\mu$-bubbles have become a central tool in positive scalar curvature geometry and have led to breakthroughs in the classification of manifolds with positive scalar curvature, the resolution of the Riemannian positive mass theorem in many settings, and the stable Bernstein program in dimensions three through five. The method is flexible: by choosing the prescription (the $\mu$-data and the function $h$), one encodes the desired curvature inequality into the Euler–Lagrange equation of the $\mu$-bubble.

Step 3: Volume and diameter estimates for $\mu$-bubbles. The third step controls the size of each $\mu$-bubble $\Sigma$. This is analogous in spirit to the dimension reduction in the Schoen–Yau proof of the positive mass theorem: one does not work directly on the full ambient manifold, but on a hypersurface of one dimension lower whose geometry is more rigid.

After Step 2, each $\mu$-bubble $\Sigma$ is a closed hypersurface in $N$ (of dimension $m-1$) lying in a thin collar between $\partial_-N$ and $\partial_+N$. Using the construction and the spectral bi-Ricci hypothesis on $N$, one shows that $\Sigma$ itself carries a spectral non-negative Ricci condition: the first eigenvalue of an operator of the form $-\Delta^\Sigma+\alpha\,\mathrm{Ric}$ is non-negative for some $\alpha\in(0,2)$ depending on $n$. One then studies manifolds with spectral non-negative Ricci curvature by methods similar to Bray’s proof of the Bishop–Gromov comparison: spectral bounds on $-\Delta+\mathrm{Ric}$ imply diameter and volume upper bounds (Theorem 5.2.13 and the discussion in §5.2.6). In particular, each component of $\partial\Omega$ has area and intrinsic diameter bounded by constants depending only on the spectral bound, not on the size of the collar.

Conclusion of the proof. Combining Steps 1–3, one obtains separating $\mu$-bubbles in long conformal annuli with uniform $\tilde g$-area and diameter bounds. As explained in the proof of the Euclidean volume-growth estimate in §5.2.6, these conformal estimates imply

\[ |B_\rho^M(0)|_g\leq C\rho^n \]

for every $\rho\gt{}0$. Thus the $\mu$-bubble argument supplies the area-growth hypothesis needed for the stable Bernstein theorems with area growth, and the cited classification results force $M$ to be an affine hyperplane. This completes the proof sketch; the conformal and spectral bi-Ricci estimates used in Steps 1–3 are proved in §5.2.1.

Conformal change of the metric on minimal hypersurfaces

Let $M^n\hookrightarrow \mathbb{R}^{n+1}$ be a complete, two-sided, stable minimal immersion with induced metric $g=\varphi^*\delta$. Fix $0\in M$ and write $r(x)=|x|$. Throughout we use the conformal metric

\[ \tilde{g}=r^{-2}g, \qquad d\tilde{\mu}=r^{-n}\,d\mu, \qquad \tilde{\nabla}=r^2\nabla^M,\quad e_i=r \tilde{e}_i. \]

We use the bi-Ricci convention fixed above: $\mathrm{BiRic}$ denotes the minimum over orthonormal two-frames, and in dimension three $\mathrm{BiRic}=\frac12 R$.

Theorem 5.2.2

Let $M^n\hookrightarrow \mathbb{R}^{n+1}$ be a complete, two-sided, stable minimal immersion with induced metric $g=\varphi^*\delta$. Then, the conformal metric $\tilde{g}=r^{-2}g$ satisfies

\[ -\tilde{\Delta}+\frac{2}{n-2}\widetilde{\mathrm{BiRic}}\geq \frac{2(-n^3+6n^{2}-4n-8)}{8(n-2)} \]

where $\tilde{\Delta}$ is the conformal Laplacian and $\widetilde{\mathrm{BiRic}}$ is the conformal bi-Ricci curvature.

Under a general conformal change $\tilde{g}=f^{-2}g$, we first relate the intrinsic Hessian on $M$ to the ambient Euclidean Hessian.

Lemma 5.2.3

Let $M^n\subset\mathbb{R}^{n+1}$ be a hypersurface with unit normal $\nu$, second fundamental form $A(X,Y)=\langle D_X Y,\nu\rangle$, and ambient connection $D$. For $f\in C^\infty(M)$ and tangent vector fields $X,Y$ on $M$,

\[ \mathrm{Hess}^M f(X,Y):=\langle \nabla^M_X\nabla^M f,Y\rangle = D^2 f(X,Y)+A(X,Y)\langle Df,\nu\rangle, \]

where $D^2 f(X,Y):=D_X(D_Y f)$ and $\nabla^M$ is the Levi-Civita connection of the induced metric.

Proof. Write the ambient gradient $Df=\nabla^M f+\langle Df,\nu\rangle\,\nu$, so $\nabla^M f$ is the tangential part. By direct computation,

\[ \begin{aligned} \nabla^M_X\nabla^M_Y f ={}& X Y (f)-\nabla_{X}^MY f = XY(f)-D_XY(f)+\langle D_X Y,\nu\rangle \langle Df,\nu\rangle\\ ={}& D^{2} f(X,Y)+A(X,Y)\langle Df,\nu\rangle. \end{aligned} \]

◻

For the conformal metric $\tilde{g}=e^{2\varphi}g$, the change of Riemannian curvatures is given by

\[ \begin{aligned} \tilde{R}_{ijij} &= e^{2\varphi} R_{ijij}-e^{2\varphi}(T_{ii}+T_{jj}) \end{aligned} \]

where

\[ T_{ij} = \nabla_i \nabla_j \varphi-\nabla_i \varphi \nabla_j \varphi+\frac{1}{2}|\nabla \varphi|^2 g_{ij}. \]

For $\varphi=-\log r$, we have

\[ T_{ii}=-\nabla^{2}_{ii}\log r - |r_i|^{2}/r^{2}+\frac{1}{2}|\nabla r|^{2}/r^{2}=-D^{2}_{ii}\log r-\frac{|r_i|^{2}}{r^{2}} +\frac{1}{2}|\nabla r|^{2}/r^{2}-A_{ii}\langle Dr,\nu\rangle/r \]

\[ T_{ii}=-\frac{1}{r^{2}}+|r_i|^{2}/r^{2}+\frac{1}{2}|\nabla r|^{2}/r^{2}-A_{ii}\langle Dr,\nu\rangle/r \]

For the conformal metric $\tilde{g}=r^{-2}g$ with $r(x)=|x|$, $\varphi=-\log r$, the conformal curvatures satisfy

\[ \begin{aligned} \tilde{R}_{\tilde{i}\tilde{j}\tilde{i}\tilde{j}} &= r^4\tilde{R}_{ijij}\\ &= r^2 R_{ijij}-r^{2}(T_{ii}+T_{jj})\\ &= r^{2}R_{ijij}-(-2+|r_i|^{2}+|r_j|^{2}+|\nabla r|^{2}-A_{ii}\langle rDr,\nu\rangle-A_{jj}\langle rDr,\nu\rangle)\\ \widetilde{\mathrm{BiRic}}(\tilde e_1,\tilde e_2) &= r^2\mathrm{BiRic}(e_1,e_2)+2(n-3)-(2n-1)|\nabla r|^{2} - (n-3)(|r_i|^{2}+|r_j|^{2})\\ &\quad +(n-3)(A_{11}+A_{22})\langle rDr,\nu\rangle, \end{aligned} \]

We compute $\mathrm{BiRic}$ on $M$ using the Gauss equation.

Proposition 5.2.4

Let $\{e_1,\ldots,e_n\}$ be a local orthonormal frame on a minimal hypersurface $M^n\subset\mathbb{R}^{n+1}$. Then

\[ \mathrm{BiRic}(e_1,e_2) =-\sum_{i=1}^n A_{1i}^2-\sum_{j=2}^n A_{2j}^2-A_{11}A_{22}. \]

Proof. Using the Gauss equation, we compute

\[ \begin{aligned} \mathrm{BiRic}(e_1,e_2) &=\sum_{i=2}^n R_{1i1i}+\sum_{j=3}^n R_{2j2j}\\ &=\sum_{i=2}^n (A_{11}A_{ii}-A_{1i}^2)+\sum_{j=3}^n (A_{22}A_{jj}-A_{2j}^2)\\ &=-\sum_{i=1}^n A_{1i}^2-\sum_{j=2}^n A_{2j}^2-A_{11}A_{22}, \end{aligned} \]

where the last line follows since $\mathrm{tr}\,A=0$ for a minimal hypersurface. ◻

Now, we choose $\tilde{e}_1$ and $\tilde{e}_2$ such that $\widetilde{\mathrm{BiRic}}(\tilde{e}_1,\tilde{e}_2)$ takes the minimum value. Then, we have

\[ \begin{aligned} \widetilde{\mathrm{BiRic}}={}& -\sum_{i=1}^n A_{1i}^2-\sum_{j=2}^n A_{2j}^2-A_{11}A_{22}+2(n-3)-(2n-1)|\nabla r|^{2} - (n-3)(|r_i|^{2}+|r_j|^{2})\\ &+(n-3)(A_{11}+A_{22})\langle rDr,\nu\rangle. \end{aligned} \]

Proposition 5.2.5

For $n \geq 3$, we have

\[ r^2 |A|^2 \geq \frac{2}{n-2} \left( (3n-3) - (2n-1)|dr|^2 - \widetilde{\mathrm{BiRic}} \right). \]

Proof. Recall that

\begin{equation} \tag{5.2.1} \label{eq:3.1} \begin{aligned} & r^2 \left( \sum_{i=1}^n A_{1i}^2 + \sum_{j=2}^n A_{2j}^2 + A_{11}A_{22} \right) + (n-3)\langle \vec x,\nu\rangle (A_{11}+A_{22}) \\ &\qquad = (4n-6) - (2n-1)|dr|^2 - (n-3)\bigl(dr(e_1)^2+dr(e_2)^2\bigr) - \widetilde{\mathrm{BiRic}} . \end{aligned} \end{equation}

Since $\langle \vec x,\nu\rangle = r\,dr(\nu)$, we use Young’s inequality to obtain

\[ \left| (n-3)\langle \vec x,\nu\rangle (A_{11}+A_{22}) \right| \leq (n-3)dr(\nu)^2 + \frac{n-3}{4}r^2(A_{11}+A_{22})^2 . \]

Combined with (5.2.1) and the fact that

\[ dr(e_1)^2+dr(e_2)^2+dr(\nu)^2 \leq 1, \]

we have

\[ \begin{aligned} & r^2 \left( \sum_{i=1}^n A_{1i}^2 + \sum_{j=2}^n A_{2j}^2 + A_{11}A_{22} + \frac{n-3}{4}(A_{11}+A_{22})^2 \right) \\ &\qquad \geq (3n-3) - (2n-1)|dr|^2 - \widetilde{\mathrm{BiRic}} . \end{aligned} \]

Now we compute, using the fact that $\operatorname{Tr} A=0$,

\[ \begin{aligned} & A_{11}^2+A_{22}^2+A_{11}A_{22} + \frac{n-3}{4}(A_{11}+A_{22})^2 \\ &\qquad = \frac12(A_{11}^2+A_{22}^2) + \frac{n-1}{4}(A_{11}+A_{22})^2 \\ &\qquad = \frac12(A_{11}^2+A_{22}^2) + \frac{n-1}{4}\sigma(A_{11}+A_{22})^2 + \frac{n-1}{4}(1-\sigma)(A_{33}+\cdots+A_{nn})^2 \\ &\qquad \leq \left( \frac12+\frac{n-1}{2}\sigma \right) (A_{11}^2+A_{22}^2) + \frac{(n-1)(n-2)}{4}(1-\sigma) (A_{33}^2+\cdots+A_{nn}^2) \\ &\qquad = \frac{n-2}{2} (A_{11}^2+\cdots+A_{nn}^2), \end{aligned} \]

where we took

\[ \sigma = \frac{n-3}{n-1} \]

in the last line. Hence, for $n\geq 3$, we have

\[ \begin{aligned} \frac{n-2}{2}r^2|A|^2 &\geq r^2 \left( \frac{n-2}{2}\sum_{i=1}^n A_{ii}^2 + \sum_{i=2}^n A_{1i}^2 + \sum_{j=3}^n A_{2j}^2 \right) \\ &\geq r^2 \left( \sum_{i=1}^n A_{1i}^2 + \sum_{j=2}^n A_{2j}^2 + A_{11}A_{22} + \frac{n-3}{4}(A_{11}+A_{22})^2 \right) \\ &\geq (3n-3) - (2n-1)|dr|^2 - \widetilde{\mathrm{BiRic}} . \end{aligned} \]

Therefore,

\[ r^2 |A|^2 \geq \frac{2}{n-2} \left( (3n-3) - (2n-1)|dr|^2 - \widetilde{\mathrm{BiRic}} \right), \]

and the proposition follows. ◻

Proposition 5.2.6

For any $\psi \in C_c^{0,1}(N,\tilde g)$, we have

\[ \int_N |\tilde \nabla \psi|_{\tilde g}^2 \, d\tilde \mu \geq \int_N \left( r^2 |A|^2 - \frac{n(n-2)}{2} + \left( \frac{n(n-2)}{2} - \frac{(n-2)^2}{4} \right) |dr|^2 \right) \psi^2 \, d\tilde \mu . \]

Proof. Recall that in the conformal metric, we have

\[ d\tilde \mu = r^{-n} d\mu \qquad\text{and}\qquad |\tilde \nabla f|_{\tilde g}^2 = r^2 |\nabla f|^2 . \]

Then the stability inequality for $M$ implies

\[ \int_N r^{n-2} |\tilde \nabla f|_{\tilde g}^2 \, d\tilde \mu \geq \int_N r^{n-2} (r^2 |A|^2) f^2 \, d\tilde \mu \]

for any $f \in C_c^{0,1}(N,\tilde g)$. We take

\[ f = r^{\frac{2-n}{2}} \psi \]

for $\psi \in C_c^{0,1}(N,\tilde g)$. Then

\[ \tilde \nabla f = r^{\frac{2-n}{2}} \tilde \nabla \psi - \frac{n-2}{2} r^{-\frac n2} \psi \tilde \nabla r , \]

\[ \begin{aligned} |\tilde \nabla f|_{\tilde g}^2 &= r^{2-n} |\tilde \nabla \psi|_{\tilde g}^2 + \frac{(n-2)^2}{4} r^{-n} \psi^2 |\tilde \nabla r|_{\tilde g}^2 \\ &\quad - (n-2) r^{1-n} \psi \langle \tilde \nabla \psi, \tilde \nabla r \rangle_{\tilde g} \\ &:= a+b+c . \end{aligned} \]

We have

\[ \int_N r^{n-2} a \, d\tilde \mu = \int_N |\tilde \nabla \psi|_{\tilde g}^2 \, d\tilde \mu . \]

Since

\[ r^{-2} |\tilde \nabla r|_{\tilde g}^2 = |dr|^2 , \]

we have

\[ \int_N r^{n-2} b \, d\tilde \mu = \int_N \frac{(n-2)^2}{4} |dr|^2 \psi^2 \, d\tilde \mu . \]

Finally, we use integration by parts and the displayed formula for $\tilde\Delta\log r$ below to compute

\[ \begin{aligned} \int_N r^{n-2} c \, d\tilde \mu &= - \int_N \frac{n-2}{2} \left\langle \tilde \nabla(\psi^2), \tilde \nabla(\log r) \right\rangle_{\tilde g} d\tilde \mu \\ &= \int_N \frac{n-2}{2} \tilde \Delta(\log r) \psi^2 \, d\tilde \mu \\ &= \int_N \left( \frac{n(n-2)}{2} - \frac{n(n-2)}{2} |dr|^2 \right) \psi^2 \, d\tilde \mu . \end{aligned} \]

Recall that we have the following

\[ \begin{aligned} \tilde{\Delta} \log r = r^{2}(\Delta \log r -(n-2)r^{-2}|\nabla r|^{2}) = n-nr^{-2}|\nabla r|^{2} \end{aligned} \]

Altogether, we obtain

as desired. ◻

Proposition 5.2.7

For $3\leq n\leq5$, we have

\[ -\tilde{\Delta}+\frac{2}{n-2}\widetilde{\mathrm{BiRic}}\geq \frac{2(-n^3+6n^{2}-4n-8)}{8(n-2)} \]

in the spectral sense.

Proof. Combining Proposition 5.2.6 with Proposition 5.2.5, for every $\psi\in C_c^{0,1}(N,\tilde g)$ we get

\[ \begin{aligned} \int_N \left( |\tilde\nabla\psi|_{\tilde g}^2 +\frac{2}{n-2}\widetilde{\mathrm{BiRic}}\psi^2 \right)d\tilde\mu \geq \int_N \left(C_0+C_1|dr|^2\right)\psi^2\,d\tilde\mu, \end{aligned} \]

where

\[ C_0=\frac{2(3n-3)}{n-2}-\frac{n(n-2)}{2}, \qquad C_1=-\frac{2(2n-1)}{n-2} +\frac{n(n-2)}{2} -\frac{(n-2)^2}{4}. \]

Since $0\leq |dr|^2\leq1$ and $C_1\leq0$ for $3\leq n\leq5$, the right-hand side is bounded from below by

\[ C_0+C_1 = \frac{2(-n^3+6n^{2}-4n-8)}{8(n-2)}. \]

This proves the claimed spectral lower bound. ◻

Construction of the $\mu$-Bubble

Theorem 5.2.8

Suppose $(N^n,g)$, $3\leq n\leq4$, is a compact manifold with boundary $\partial N=\partial_+N\cup\partial_-N$. Assume that there are $0\lt{}a\leq 2$ and a positive smooth function $w$ such that

\[ -a\Delta^N w+\mathrm{BiRic}_N\,w\geq w. \]

Suppose that the distance between $\partial_-N$ and $\partial_+N$ is bounded below by $5\pi$. Then one can find a smooth $\mu$-bubble $\Sigma$ separating $\partial_-N$ from $\partial_+N$ such that, for every $\psi\in C_c^\infty(\Sigma)$,

\[ \frac{4}{4-a}\int_\Sigma |\nabla^\Sigma\psi|^2 \geq \int_\Sigma \left(\frac12-\mathrm{Ric}_\Sigma\right)\psi^2. \]

Equivalently, in the spectral sense,

\[ -\frac{4}{4-a}\Delta^\Sigma+\mathrm{Ric}_\Sigma\geq \frac12. \]

The constant $1$ is only a normalization and can be replaced by any positive constant after rescaling the metric.

Second variation of weighted area functional

Lemma 5.2.9

The minimizer to the following functional

\[ \mathcal{A}(\Omega)=\int_{\partial \Omega} w d\mathcal{H}^{n-1}-\int_{\Omega} h w d\mathcal{H}^{n}, \]

satisfies the following equation

\[ \begin{aligned} {} & \int_{ \Sigma} w \left|\nabla^\Sigma \phi\right|^2+\frac{1}{2}R_\Sigma\phi^2w-\phi\Delta^\Sigma w-\frac{1}{2}w^{-1}\left< \nabla^N w, \nu \right> ^2\phi^2-\int_{ \Sigma} \phi^2(-\Delta^Nw+\frac{1}{2}R_Nw) \\ {} & -\int_{ \Sigma}\left[ \phi^2 w \left< \nabla^Nh,\nu \right> + \frac{1}{2}h^2\phi^2w \right]\geq 0. \end{aligned} \]

for any smooth function $\phi$ on $\Sigma$.

Given a domain $\Omega$ in $N$, we do a variation $\Omega_t$ under vector field $V$ and denote $\Sigma_t=\partial\Omega_t\backslash \partial_-N$ and $\Sigma=\Sigma_0$. We choose $V$ such that $\nabla_{V}V=0$ and $V=\phi \nu$ along $\Sigma$. (Note that in general, we do not have $V\bot \Sigma_t$ for every $t$.)

Lemma 5.2.10

The following first and second variation formulas hold:

\[ \begin{aligned} \mathcal{A}'={} & \int_{ \Sigma} \phi\left[ \left< \nabla^Nw, \nu\right>+w H-wh \right], \text{ Recall that }H=\sum_{i =1}^{n}\left< \nabla_{e_i}e_i, \nu \right>\\ \mathcal{A}''={} & \int_{ \Sigma} \phi^2 (\Delta^N w-\Delta^\Sigma w)+\int_{ \Sigma} w\left( \left|\nabla^\Sigma \phi\right|^2-\left( |A_\Sigma|^2+\mathrm{Ric}(\nu,\nu) \right)\phi^2 \right) \\ {}& - \int_{ \Sigma}\left(\phi^2 h\left< \nabla^N w,\nu \right>+\phi^2 w\left< \nabla^Nh,\nu \right>\right). \end{aligned} \]

The first variation is straightforward.

\[ \begin{aligned} \mathcal{A}'={} & \int_{ \Sigma} \left< \nabla^Nw, V \right>-w \mathrm{div}^\Sigma(V)-hw \left< V, \nu \right> d\mathcal{H}^3 \\ ={} & \int_{ \Sigma} \left< \nabla^Nw, V \right>+w \mathrm{div}^\Sigma(V^T)+w H \left< V, \nu \right> -hw \left< V,\nu \right> d\mathcal{H}^{n-1} \end{aligned} \]

Hence, the stationary gives us the formula

\[ H=h-w^{-1}\left< \nabla^N w,\nu \right> \]

For the derivative of $\int_{\Sigma} w\,\mathrm{div}^\Sigma(V^T)\,d\mathcal{H}^{n-1}$, we rewrite it using integration by parts as

\[ -\int_{ \Sigma} \left< \nabla^\Sigma w, V^T \right>. \]

After differentiating,

\[ -\int_{ \Sigma} \left< \nabla^\Sigma w, \nabla_{V}(V^T) \right> =\int_{ \Sigma} \left< \nabla^\Sigma w, \phi \nabla_{V}\nu \right> \]

where we have used $\nabla_{V}V=0$. Note that we have

\[ \left< \nabla_{V}\nu,e_i \right> = -\left<\nu, \nabla_{V}e_i \right> = -\left< \nu, \nabla_{e_i}(\phi\nu) \right> =-e_i(\phi) \implies \nabla_{V}\nu=-\nabla^\Sigma \phi \]

Then, the derivative of $\int_{ \Sigma} \left< \nabla^\Sigma w, \phi \nabla_{V}\nu \right>$ is

\[ \int_{ \Sigma} \left< \nabla^\Sigma w, \phi \nabla_{V}\nu \right> = \int_{ \Sigma} -\left< \nabla^\Sigma w, \nabla^\Sigma\phi \right> \phi \]

Now, we can compute the second variation. Note that we can use $V\left< V, \nu \right> =0$.

\[ \begin{aligned} \mathcal{A}''={}& \int_{ \Sigma} \phi^2 \mathrm{Hess}^Nw(\nu,\nu)-\phi \left< \nabla^\Sigma w, \nabla^\Sigma \phi \right> -\phi^2 \left< \nabla^Nw, \nu \right> H \\ {}& -\int_{ \Sigma} w\phi \left( \Delta^\Sigma \phi +(|A_\Sigma|^2+\mathrm{Ric}(\nu,\nu))\phi \right) -\int_{ \Sigma}\left(\phi^2 h\left< \nabla^N w,\nu \right>+\phi^2w\left< \nabla^Nh,\nu \right>\right). \end{aligned} \]

For the Hessian, we have

\[ \begin{aligned} \mathrm{Hess}^Nw(\nu,\nu)={}&\mathrm{div}^N(\nabla^Nw)-\mathrm{div}^\Sigma(\nabla^Nw)=\Delta^Nw -\Delta^\Sigma w - \mathrm{div}^\Sigma(\left< \nabla^Nw, \nu \right> \nu)\\ ={}&\Delta^Nw-\Delta^\Sigma w+\left< \nabla^Nw,\nu \right>H \end{aligned} \]

using integration by parts, we have

\[ \begin{aligned} &\mathcal{A}''=\int_{ \Sigma} \phi^{2}(\Delta^N w-\Delta^\Sigma w)+ w\left( |\nabla^\Sigma \phi|^{2}-(|A|^{2}+\mathrm{Ric}_N(\nu,\nu))\phi^{2} \right)\\ &-\int_{ \Sigma}\left(\phi^{2}h\left< \nabla^N w,\nu \right>+\phi^{2} w \left< \nabla^Nh,\nu \right>\right). \end{aligned} \]

This is the weighted-area second variation formula used below.

Properties of the minimizer

Proposition 5.2.11

Suppose $\Omega$ is the minimizer to $\mathcal{A}$. Assume that the weight is $W=w^a$, where $0\lt{}a\leq 2$, $w\gt{}0$, and

\[ -a\Delta^Nw+\mathrm{BiRic}_N\,w\geq w. \]

We also assume that the function $h$ satisfies the following condition:

\[ 1+h^{2}-2|\nabla^N h|\geq 0. \]

Then, for any $\psi\in C_c^\infty(\Sigma)$, we have

\[ \begin{aligned} \frac{4}{4-a}\int_{ \Sigma} |\nabla^\Sigma \psi|^{2}\geq{}& \int_{ \Sigma} \psi^{2}\left( \frac{1}{2}-Ric_\Sigma \right) \end{aligned} \]

We can also write it as

\[ -\frac{4}{4-a}\Delta^\Sigma+Ric_\Sigma\geq \frac{1}{2}. \]

Proof. Suppose $\Omega$ is the minimizer to $\mathcal{A}$. In the spectral bi-Ricci application the weight in the functional is

\[ W=w^a, \]

and we consider

\[ \phi=W^{-\frac12}\psi=w^{-\frac a2}\psi. \]

Then

\[ \nabla^\Sigma\phi = W^{-\frac12}\nabla^\Sigma\psi -\frac12 W^{-\frac12}\psi\nabla^\Sigma\log W. \]

Hence

\[ \begin{aligned} \int_{\Sigma} W|\nabla^\Sigma\phi|^2 = \int_{\Sigma}|\nabla^\Sigma\psi|^2 +\frac14\psi^2|\nabla^\Sigma\log W|^2 -\psi\left<\nabla^\Sigma\psi,\nabla^\Sigma\log W\right>. \end{aligned} \]

The first variation gives

\[ H=h-\left<\nabla^N\log W,\nu\right> =h-a\left<\nabla^N\log w,\nu\right>. \]

Put

\[ v=\left<\nabla^N\log w,\nu\right>. \]

Then $H=h-av$, and hence

\[ ahv=\frac12h^2+\frac12a^2v^2-\frac12H^2. \]

We also need to compare the ambient and intrinsic Laplacians of the weight $W=w^a$. Along $\Sigma$,

\[ \begin{aligned} \frac{\Delta^\Sigma W}{W} ={}& a\frac{\Delta^\Sigma w}{w} +a(a-1)|\nabla^\Sigma\log w|^2,\\ \frac{\Delta^N W}{W} ={}& a\frac{\Delta^N w}{w} +a(a-1)\left(|\nabla^\Sigma\log w|^2+v^2\right). \end{aligned} \]

Thus

\[ \frac{\Delta^N W}{W}-\frac{\Delta^\Sigma W}{W} = a\left(\frac{\Delta^N w}{w}-\frac{\Delta^\Sigma w}{w}\right) +a(a-1)v^2. \]

Since $\Sigma$ is stable for the weighted functional, the second variation formula with weight $W$ gives, after moving the potential terms to the right-hand side,

\begin{equation} \tag{5.2.2} \begin{aligned} &\int_\Sigma W\left(|\nabla^\Sigma\phi|^2 -a w^{-1}\Delta^\Sigma w\,\phi^2\right) \notag\\ \geq{}& \int_\Sigma \psi^2\left( -a\frac{\Delta^N w}{w} +|A|^2+\mathrm{Ric}_N(\nu,\nu)-\frac12H^2 +\frac12h^2+\left<\nabla^Nh,\nu\right> \right) \notag\\ &+\frac{a(2-a)}2\int_\Sigma \psi^2 v^2. \label{eq:mububble-weighted-stability-with-normal-term} \end{aligned} \end{equation}

The final term is obtained by combining the Laplacian comparison contribution $-a(a-1)v^2$ with the $ahv$ term. Since $0\lt{}a\leq2$, it is nonnegative and can be discarded.

It remains to estimate the left-hand side of (5.2.2). After substituting $\phi=w^{-a/2}\psi$, it becomes

\[ \begin{aligned} \int_\Sigma |\nabla^\Sigma\psi|^2 +a\psi\left<\nabla^\Sigma\psi,\nabla^\Sigma\log w\right> -\left(a-\frac{a^2}{4}\right)\psi^2|\nabla^\Sigma\log w|^2, \end{aligned} \]

and Young’s inequality with $\varepsilon=(4-a)^{-1}$ gives the estimate. Therefore,

\[ \begin{aligned} \int_{\Sigma} W\left(|\nabla^\Sigma\phi|^2-a w^{-1}\Delta^\Sigma w\,\phi^2\right) \leq \frac{4}{4-a}\int_{\Sigma}|\nabla^\Sigma\psi|^2. \end{aligned} \]

Combining this estimate with (5.2.2) and the spectral inequality $-a\Delta^Nw+\mathrm{BiRic}_Nw\geq w$ gives

\[ \begin{aligned} \frac{4}{4-a}\int_{\Sigma}|\nabla^\Sigma\psi|^2 \geq{}& \int_\Sigma \psi^2\left( 1-\mathrm{BiRic}_N +|A|^2+\mathrm{Ric}_N(\nu,\nu)-\frac12H^2 \right) \\ &+\int_\Sigma \psi^2\left(\frac12h^2+\left<\nabla^Nh,\nu\right>\right). \end{aligned} \]

Now, we need to use $|A|^{2}+\mathrm{Ric}_N(\nu,\nu)$ to bound $\mathrm{BiRic}_N$.

Lemma 5.2.12

We have the following inequality:

\[ |A|^{2}+\mathrm{Ric}_N(\nu,\nu)\geq \mathrm{BiRic}_N-\mathrm{Ric}_\Sigma + \frac{6-n}{4}H^{2}. \]

This is the extension of the usual Schoen–Yau trick. Recall that we have (for $n=3$)

\[ |A|^{2}+\mathrm{Ric}_N(\nu,\nu)=\frac{1}{2}R_N-\frac{1}{2}R_\Sigma+\frac{1}{2}|A|^2+\frac{1}{2}H^2\geq \frac{1}{2}R_N-\frac{1}{2}R_\Sigma+\frac{3}{4}H^{2}. \]

This is exactly the same as the inequality in the lemma.

Proof. Suppose $e_1\in T_p\Sigma$ is a unit direction where $Ric_\Sigma$ attains its minimum. Using the Gauss equation, we have

\[ \begin{aligned} Ric_\Sigma(e_1,e_1) ={}&\sum_{i=2}^{n-1}R^\Sigma_{1i1i} \\ ={}&\sum_{i=2}^{n-1}R^N_{1i1i} +A_{11}\sum_{i=2}^{n-1}A_{ii} -\sum_{i=2}^{n-1}A_{1i}^{2}. \end{aligned} \]

Note that we have

\[ \begin{aligned} A_{11}\sum_{i=2}^{n-1}A_{ii}={}& -A_{11}^{2}+A_{11}H=-A_{11}^{2}-H\sum_{i=2}^{n-1}A_{ii}+H^{2}\\ \geq{}& -A_{11}^{2}-\frac{1}{n-2}\left( \sum_{i=2}^{n-1}A_{ii} \right)^{2}+\left( 1-\frac{n-2}{4} \right)H^{2}\\ \geq{}& -\sum_{i=1}^{n-1}A_{ii}^{2}+\frac{6-n}{4}H^{2} \end{aligned} \]

By the choice of $e_1$, $\mathrm{Ric}_\Sigma=\mathrm{Ric}_\Sigma(e_1,e_1)$. Hence, we have

\[ \begin{aligned} \mathrm{Ric}_\Sigma+\mathrm{Ric}_N(\nu,\nu) \geq{}& \sum_{i=2}^{n-1}R^N_{1i1i} +\mathrm{Ric}_N(\nu,\nu)-|A|^{2}+\frac{6-n}{4}H^{2}\\ \geq{}& \mathrm{BiRic}_N-|A|^{2} + \frac{6-n}{4}H^{2}. \end{aligned} \]

Equivalently,

\[ |A|^{2}+\mathrm{Ric}_N(\nu,\nu) \geq \mathrm{BiRic}_N-\mathrm{Ric}_\Sigma+\frac{6-n}{4}H^{2}, \]

which is the desired inequality. ◻

Now, we go back to our inequality. When $3\leq n\leq 4$, we have $\frac{6-n}{4}\geq \frac{1}{2}$ and therefore

\[ \begin{aligned} \frac{4}{4-a}\int_{ \Sigma} |\nabla^\Sigma \psi|^{2}\geq{}& \int_{ \Sigma} \psi^{2}(1-Ric_\Sigma) +\int_\Sigma\psi^2\left(\frac{1}{2}h^{2}+\left< \nabla^Nh,\nu \right>\right). \end{aligned} \]

Together with the condition for $h$, $1+h^{2}-2|\nabla^N h|\geq0$, we have

\[ 1+\frac12h^2+\left<\nabla^Nh,\nu\right> \geq \frac12+\frac12(1+h^2-2|\nabla^Nh|) \geq \frac12. \]

Hence

\[ \frac{4}{4-a}\int_\Sigma|\nabla^\Sigma\psi|^2 \geq \int_{ \Sigma} \psi^{2}\left( \frac{1}{2}-Ric_\Sigma \right). \]

This is the desired inequality. ◻

Completion of the $\mu$-Bubble Construction

To prove Theorem 5.2.8, choose $h$ by

\[ h(x)=-\frac{1}{2}\tan(\frac{\tilde{d}(x,\partial_-N)}{4}-\frac{\pi}{2}) \]

where $\tilde{d}(x,\partial_-N)$ is a smoothing of $d_N(x,\partial_-N)$ with $\mathrm{Lip}\,\tilde{d}\leq 2$. We denote $\varphi(x)=\frac{\tilde{d}(x,\partial_-N)}{4}-\frac{\pi}{2}$. Then

\[ |\nabla^N h| \leq \frac12(1+\tan^2\varphi), \qquad 2|\nabla^N h|\leq 1+h^2 . \]

To show the existence of the minimizer, fix a domain $\Omega_0$ containing $\partial_-N$ but not $\partial_+N$ such that $\partial \Omega_0$ lies in $\tilde{d}(x,\partial_-N)\lt{} 4\pi$. We consider minimizing the following (relative) energy functional

\[ \mathcal{A}(\Omega)=\int_{ \partial \Omega} w^a d\mathcal{H}^{n-1}-\int_{\Omega\backslash \Omega_0} h w^a d\mathcal{H}^{n}+\int_{ \Omega_0\backslash \Omega}hw^a d\mathcal{H}^n \]

The direct method gives a minimizer in the corresponding relative homology class. Since $3\leq n\leq4$, the free boundary part $\Sigma=\partial\Omega\setminus\partial_-N$ is smooth after the usual regularity theory for prescribed-mean-curvature hypersurfaces. The choice $d_N(\partial_+N,\partial_-N)\geq5\pi$ gives enough room for the smoothing $\tilde d$ and hence for the above choice of $h$. Applying the stability inequality from the preceding proposition proves Theorem 5.2.8.

Diameter estimate and volume estimate under spectral Ricci curvature bound

The following result of Antonelli–Xu [AX24] gives the radius/diameter and volume estimates under a spectral Ricci lower bound. In this subsection $\mathrm{Ric}_N(v,v)$ denotes the Ricci tensor on a unit vector; when $\mathrm{Ric}_N$ appears without arguments in a spectral inequality, it means the smallest eigenvalue of the Ricci tensor. The bi-Ricci notation is the one fixed above. In the low-dimensional range considered here, the minimizers appearing in the $\mu$-bubble arguments are smooth. The parameter $\alpha$ below denotes the coefficient of the spectral Laplacian term.

Theorem 5.2.13

Let $(N^n,g)$ be a compact smooth Riemannian manifold, $3\leq n\leq 5$, and let

\[ 0\leq \alpha\leq \frac{n-1}{n-2},\qquad \lambda\gt{}0 . \]

Suppose that there is a positive smooth function $u$ such that

\begin{equation} \tag{5.2.3} \label{eq:AX-spectral-Ric} -\alpha\Delta u+\mathrm{Ric}_N\,u\geq (n-1)\lambda u . \end{equation}

Let $(\widetilde N,\tilde g)$ be the universal cover and let $\tilde u$ be the lift of $u$. Then the diameter upper bound is

\begin{equation} \tag{5.2.4} \label{eq:AX-diam-bound} \mathrm{diam}(\widetilde N,\tilde g) \leq \frac{\pi}{\sqrt{\lambda}} \left(\frac{\max_N u}{\min_N u}\right)^{\frac{n-3}{n-1}\alpha}. \end{equation}

Moreover the sharp volume upper bound is

\begin{equation} \tag{5.2.5} \label{eq:AX-volume-bound} |\widetilde N| \leq \lambda^{-\frac n2}|S^n|, \end{equation}

where $S^n$ denotes the unit round sphere. In particular $\pi_1(N)$ is finite. If equality holds in the volume estimate, then $\tilde u$ is constant and $\widetilde N$ is the round sphere of radius $\lambda^{-1/2}$.

Corollary 5.2.14

Let $(N^n,g)$ be complete, not assumed compact, with $3\leq n\leq 5$. If $n\gt{}3$ and $0\leq\alpha\lt{}\frac{4}{n-1}$, or if $n=3$ and $0\leq\alpha\leq2$, and if (5.2.3) holds for some positive smooth $u$, then $N$ is compact. Moreover $\pi_1(N)$ is finite and, for the universal cover $\widetilde N$,

\[ \mathrm{diam}(\widetilde N,\tilde g) \leq \frac{\pi}{\sqrt{\lambda}} \left(\frac{\max_N u}{\min_N u}\right)^{\frac{n-3}{n-1}\alpha}, \qquad |\widetilde N|\leq \lambda^{-\frac n2}|S^n|. \]

In particular $|N|\leq \lambda^{-n/2}|S^n|$. In addition there is a constant $C=C(n,\alpha)$ such that

\[ \mathrm{diam}(N)\leq C\lambda^{-1/2}. \]

Proof. This is precisely the complete-case corollary in [AX24], based on Xu’s subcritical spectral Bonnet–Myers theorem. The latter gives compactness and the uniform bound $\mathrm{diam}(N)\leq C(n,\alpha)\lambda^{-1/2}$ in the stated range. Once compactness is known, Theorem 5.2.13 applies and gives (5.2.4) and (5.2.5). Since $N$ is the quotient of $\widetilde N$ by a finite group of deck transformations, the same volume upper bound also holds for $N$. ◻

Even in the larger sharp range $0\leq\alpha\leq (n-1)/(n-2)$, if $N$ is complete and the positive function $u$ in (5.2.3) satisfies $0\lt{}\inf_N u\leq\sup_Nu\lt{}\infty$, then Corollary 5.2.17 below proves compactness directly. Thus noncompact examples in the super-subcritical range must have $u$ degenerating at infinity.

Weighted geodesic proof of the diameter estimates.

We also record the weighted-geodesic calculation, since it gives another way to see the Bonnet–Myers part.

For a positive function $u$ on $(M^m,g)$, we define the weighted geodesic distance from $p$ to $q$ by

\[ L_u^\alpha(p,q) = \inf_\eta\int_\eta u^\alpha\,ds_g , \]

where the infimum is taken over piecewise smooth curves from $p$ to $q$. As above, $\mathrm{Ric}_M$ in a spectral inequality denotes the smallest eigenvalue of the Ricci tensor, while $\mathrm{Ric}_M(T,T)$ denotes the tensor on the vector $T$.

Lemma 5.2.15

Let $(M^m,g)$ be complete, $m\geq3$, and suppose that $u\gt{}0$ satisfies

\begin{equation} \tag{5.2.6} \label{eq:AX-alpha-spectral-positive} -\alpha\Delta u+\mathrm{Ric}_M\,u\geq (m-1)\lambda u, \qquad \lambda\gt{}0 . \end{equation}

Let $\alpha:[0,l]\to M$ be an $L_u^\alpha$-minimizing curve from $p$ to $q$, parametrized by $g$-arclength, and assume first that $q$ is not a weighted cut point along $\alpha$. Then the following hold in the barrier sense at $q$.

If $0\leq\alpha\lt{}4/(m-1)$, then for every $C^1$ function $\psi$ on $[0,l]$ with

\[ \psi(0)=0,\qquad \psi(l)=u(q)^{\alpha/2}, \]

we have

\begin{equation} \tag{5.2.7} \label{eq:AX-subcritical-weighted-lap} \Delta_q L_u^\alpha(p,q) \leq \int_0^l \left(C_{m,\alpha}\psi_s^2-(m-1)\lambda\psi^2\right)\,ds, \end{equation}

where

\[ C_{m,\alpha} = (m-1)+ \frac{\alpha(m-3)^2}{4\left(1-\frac{m-1}{4}\alpha\right)} . \]

In particular, choosing

\[ \psi(s)=u(q)^{\alpha/2}\sin\left(\frac{\pi s}{2l}\right) \]

gives

\begin{equation} \tag{5.2.8} \label{eq:AX-subcritical-weighted-lap-sine} \Delta_q L_u^\alpha(p,q) \leq u(q)^\alpha\left( \frac{C_{m,\alpha}\pi^2}{8l} -\frac{(m-1)\lambda l}{2} \right). \end{equation}

If $0\leq\alpha\leq (m-1)/(m-2)$, then for every $C^1$ function $\psi$ with

\[ \psi(0)=0,\qquad \psi(l)=u(q)^{\alpha/(m-1)}, \]

we have

\begin{equation} \tag{5.2.9} \label{eq:AX-bounded-weighted-lap} \Delta_q L_u^\alpha(p,q) \leq \int_0^l u^{\frac{m-3}{m-1}\alpha} \left((m-1)\psi_s^2-(m-1)\lambda\psi^2\right)\,ds . \end{equation}

Proof. Let $T=\alpha'$ and put

\[ \tilde g=u^{2\alpha}g . \]

Then $L_u^\alpha$ is exactly the distance function of the conformal metric $\tilde g$. Write

\[ \rho=L_u^\alpha(p,\cdot). \]

Along $\alpha$ we have, by the definition of $\tilde g$ and of the distance function $\rho$,

\[ d\tilde s=u^\alpha\,ds,\qquad \tilde T=u^{-\alpha}T, \qquad \nabla^g L_u^\alpha=u^\alpha T . \]

We first compute $\tilde\Delta\rho$ in the conformal metric. Choose a $\tilde g$-parallel orthonormal frame along $\alpha$,

\[ \tilde E_1,\ldots,\tilde E_{m-1},\qquad \tilde E_m=\tilde T, \]

and write $\tilde E_i=u^{-\alpha}e_i$. Thus $e_1,\ldots,e_{m-1},T$ is $g$-orthonormal and $e_i\perp T$. Let $\phi$ be a test function with $\phi(0)=0$ and $\phi(l)=1$, and set

\[ \tilde V_i=\phi\tilde E_i . \]

These fields vanish at $p$ and equal $\tilde E_i(q)$ at $q$. Hence the ordinary second variation formula for the $\tilde g$-distance gives, for $1\leq i\leq m-1$,

\[ \tilde\nabla^2\rho(\tilde E_i,\tilde E_i)(q) \leq \int_0^{\tilde l} \left( (\tilde T\phi)^2 -\phi^2\widetilde R(\tilde T,\tilde E_i,\tilde T,\tilde E_i) \right)\,d\tilde s . \]

The missing $m$-th direction is radial and contributes nothing, since $\tilde\nabla^2\rho(\tilde T,\tilde T)=0$ away from the cut locus. Summing the preceding inequality over the transverse directions gives

\begin{equation} \tag{5.2.10} \label{eq:AX-tilde-index-simple} \tilde\Delta\rho(q) \leq \int_0^l u^{-\alpha} \left((m-1)\phi_s^2-\phi^2\widetilde{\mathrm{Ric}}(T,T)\right)\,ds . \end{equation}

Here we used $d\tilde s=u^\alpha ds$, $\tilde T\phi=u^{-\alpha}\phi_s$, and $\widetilde{\mathrm{Ric}}(\tilde T,\tilde T)=u^{-2\alpha}\widetilde{\mathrm{Ric}}(T,T)$.

Now return to the original metric. The conformal Laplacian formula for $\tilde g=e^{2\log u^\alpha}g$ gives, at $q$,

\begin{equation} \tag{5.2.11} \label{eq:AX-lap-conformal-weighted} \Delta^g\rho = u(q)^{2\alpha}\tilde\Delta\rho-(m-2)u(q)^\alpha(\log u^\alpha)_s(l). \end{equation}

Combining (5.2.10) and (5.2.11), we get

\begin{equation} \tag{5.2.12} \begin{aligned} \Delta^g\rho(q) \leq{}& u^{2\alpha}(q)\int_0^l u^{-\alpha} \left((m-1)\phi_s^2-\phi^2\widetilde{\mathrm{Ric}}(T,T)\right)\,ds \notag\\ &\qquad -(m-2)u(q)^\alpha(\log u^\alpha)_s(l). \label{eq:AX-before-test-change} \end{aligned} \end{equation}

Now rewrite the curvature term in (5.2.12) in the original metric. The conformal Ricci formula is

\[ \begin{aligned} \widetilde{\mathrm{Ric}} ={}& \mathrm{Ric} -(m-2)\left( \nabla^2\log u^\alpha -d\log u^\alpha\otimes d\log u^\alpha \right) \\ &-\left( \Delta\log u^\alpha +(m-2)|\nabla\log u^\alpha|^2 \right)g . \end{aligned} \]

The conformal connection formula is

\[ \tilde\nabla_XY = \nabla_XY+X(\log u^\alpha)Y+Y(\log u^\alpha)X -g(X,Y)\nabla\log u^\alpha . \]

Applying it to the $\tilde g$-geodesic equation $\tilde\nabla_{\tilde T}\tilde T=0$ gives

\[ 0=u^{2\alpha}\tilde\nabla_{\tilde T}\tilde T = \nabla_TT+(\log u^\alpha)_sT-\nabla\log u^\alpha, \]

and therefore

\[ \nabla_TT=\nabla^\perp\log u^\alpha,\qquad \nabla^2\log u^\alpha(T,T) =(\log u^\alpha)_{ss}-|\nabla^\perp\log u^\alpha|^2 . \]

Using this in the Ricci formula gives the pointwise identity along $\alpha$

\begin{equation} \tag{5.2.13} \begin{aligned} \widetilde{\mathrm{Ric}}(T,T) ={}& \mathrm{Ric}_M(T,T) -(m-2)\left( (\log u^\alpha)_{ss}-\nabla_TT \cdot \nabla \log u^\alpha -(\log u^\alpha)_s^2 \right) \notag\\ &-\left( \Delta\log u^\alpha +(m-2)|\nabla\log u^\alpha|^2 \right) \notag\\ ={}& \mathrm{Ric}_M(T,T) -(m-2)\left( (\log u^\alpha)_{ss}-|\nabla^\perp\log u^\alpha|^2 -(\log u^\alpha)_s^2 \right) \notag\\ &-\left(\Delta\log u^\alpha +(m-2)|\nabla\log u^\alpha|^2\right) \notag\\ ={}& \mathrm{Ric}_M(T,T) -\Delta\log u^\alpha -(m-2)(\log u^\alpha)_{ss}. \label{eq:AX-tilde-ric-to-g} \end{aligned} \end{equation}

The endpoint term in (5.2.12) is written in the same $u(q)^{2\alpha}\int u^{-\alpha}(\cdots)\,ds$ scale as

\[ -(m-2)u(q)^\alpha(\log u^\alpha)_s(l) = -u(q)^{2\alpha}\int_0^l \left((m-2)\phi^2u^{-\alpha}(\log u^\alpha)_s\right)_s\,ds, \]

because $\phi(0)=0$ and $\phi(l)=1$. Substituting (5.2.13) into (5.2.12) and expanding this total derivative gives the original-metric formula

\begin{equation} \tag{5.2.14} \begin{aligned} \Delta^g\rho(q) \leq{}& u(q)^{2\alpha}\int_0^l u^{-\alpha}\Big[ (m-1)\phi_s^2 -2(m-2)\phi\phi_s(\log u^\alpha)_s \notag\\ &\qquad +(m-2)\phi^2(\log u^\alpha)_s^2 -\mathrm{Ric}_M(T,T)\phi^2 +\phi^2\Delta\log u^\alpha \Big]\,ds .\label{eq:AX-original-metric-before-change} \end{aligned} \end{equation}

Using (5.2.6) and $\mathrm{Ric}_M(T,T)\geq \mathrm{Ric}_M$, along the curve

\[ -\mathrm{Ric}_M(T,T)+\alpha\frac{\Delta u}{u} \leq -(m-1)\lambda . \]

Together with

\[ \Delta\log u^\alpha = \alpha\frac{\Delta u}{u} -\alpha\frac{|\nabla u|^2}{u^2}, \]

this gives

\begin{equation} \tag{5.2.15} \begin{aligned} \Delta^g\rho(q) \leq{}& u(q)^{2\alpha}\int_0^l u^{-\alpha}\Big[ (m-1)\phi_s^2 -2(m-2)\phi\phi_s(\log u^\alpha)_s \notag\\ &\qquad +(m-2)\phi^2(\log u^\alpha)_s^2 -(m-1)\lambda\phi^2 -\alpha\phi^2\frac{|\nabla u|^2}{u^2} \Big]\,ds .\label{eq:AX-original-metric-lambda-before-change} \end{aligned} \end{equation}

Only at this point do we change the test function. The subcritical estimate uses the substitution

\[ \phi=\frac{u^{\alpha/2}}{u(q)^\alpha}\psi,\qquad \psi(0)=0,\qquad \psi(l)=u(q)^{\alpha/2}. \]

Thus

\[ \phi_s=\frac{u^{\alpha/2}}{u(q)^\alpha} \left(\psi_s+\frac{\alpha}{2}\psi\frac{u_s}{u}\right), \qquad u_s=T(u). \]

Inserting this into (5.2.15) cancels the weight $u^\alpha$ and gives

\begin{equation} \tag{5.2.16} \begin{aligned} \Delta_q L_u^\alpha(p,q) \leq{}& \int_0^l\Big[ (m-1)\psi_s^2 +\alpha(3-m)\psi\psi_s\frac{u_s}{u} +\frac{m-1}{4}\alpha^2\psi^2\frac{u_s^2}{u^2} \notag\\ &\qquad -\alpha\psi^2\frac{|\nabla u|^2}{u^2} -(m-1)\lambda\psi^2 \Big]\,ds .\label{eq:AX-half-power-change} \end{aligned} \end{equation}

Since $|\nabla u|^2\geq u_s^2$, this implies

\[ \begin{aligned} \Delta_q L_u^\alpha(p,q) \leq{}& \int_0^l\Big[ (m-1)\psi_s^2 +\alpha(3-m)\psi\psi_s\frac{u_s}{u} \notag\\ &\qquad -\alpha\left(1-\frac{m-1}{4}\alpha\right) \psi^2\frac{u_s^2}{u^2} -(m-1)\lambda\psi^2 \Big]\,ds . \end{aligned} \]

When $\alpha\lt{}4/(m-1)$, Cauchy’s inequality gives

\[ \alpha(3-m)\psi\psi_s\frac{u_s}{u} -\alpha\left(1-\frac{m-1}{4}\alpha\right)\psi^2\frac{u_s^2}{u^2} \leq \frac{\alpha(m-3)^2}{4\left(1-\frac{m-1}{4}\alpha\right)}\psi_s^2 . \]

This proves (5.2.7); the sine choice gives (5.2.8).

For the bounded-$u$ diameter estimate one uses a different exponent. Put

\[ \phi = \frac{u^{\frac{m-2}{m-1}\alpha}}{u(q)^\alpha}\psi, \qquad \psi(0)=0,\qquad \psi(l)=u(q)^{\alpha/(m-1)} . \]

Then

\[ \phi_s = \frac{u^{\frac{m-2}{m-1}\alpha}}{u(q)^\alpha} \left( \psi_s+\frac{m-2}{m-1}\alpha\psi\frac{u_s}{u} \right). \]

Substituting this into (5.2.15), the cross term cancels exactly and we obtain

\[ \begin{aligned} \Delta_q L_u^\alpha(p,q) \leq{}& \int_0^l u^{\frac{m-3}{m-1}\alpha}\Big[ (m-1)\psi_s^2 +\alpha\left(\frac{m-2}{m-1}\alpha-1\right) \psi^2\frac{u_s^2}{u^2} \notag\\ &\qquad -\alpha\psi^2\frac{|\nabla u|^2-u_s^2}{u^2} -(m-1)\lambda\psi^2 \Big]\,ds . \end{aligned} \]

If $\alpha\leq (m-1)/(m-2)$, the two terms involving derivatives of $u$ are nonpositive. Dropping them gives (5.2.9). ◻

Corollary 5.2.16

Let $(M^m,g)$ be complete, $m\geq3$, and suppose $u\gt{}0$ satisfies (5.2.6). If

\[ 0\leq\alpha\lt{}\frac{4}{m-1}, \]

then

\begin{equation} \tag{5.2.17} \label{eq:AX-subcritical-diam-from-geodesic} \mathrm{diam}(M) \leq \pi\sqrt{\frac{C_{m,\alpha}}{(m-1)\lambda}} . \end{equation}

In particular $M$ is compact.

Proof. Assume first that two points $p,q$ are joined by an $L_u^\alpha$-minimizing curve $\alpha:[0,l]\to M$ parametrized by $g$-arclength. Let $x=\alpha(l/2)$. The weighted excess

\[ e(y)=L_u^\alpha(p,y)+L_u^\alpha(y,q)-L_u^\alpha(p,q) \]

has a local minimum at $x$. Using the two subsegments of $\alpha$ as barriers and applying (5.2.8) to each half, we get, in the viscosity sense,

\[ 0\leq\Delta e(x) \leq u(x)^\alpha \left( \frac{C_{m,\alpha}\pi^2}{2l} -\frac{(m-1)\lambda l}{2} \right). \]

Therefore $l\leq \pi\sqrt{C_{m,\alpha}/((m-1)\lambda)}$. Since $d_g(p,q)\leq l$, this gives (5.2.17).

If the weighted minimizer is not realized, take a minimizing sequence. The usual Calabi limiting argument gives a broken minimizing object made of finite minimizing segments, weighted minimizing rays, and weighted minimizing lines. At the point where the corresponding excess vanishes, replace any ray by the point at parameter $t$ on that ray and use the finite-segment barrier above. The right hand side contains the term $-(m-1)\lambda t/2$ for that ray and is negative for $t$ sufficiently large, contradicting the local minimum of the excess. Thus the non-realized case cannot occur, and the same diameter bound holds for all pairs of points. Hopf–Rinow then implies compactness. ◻

Corollary 5.2.17

Let $(M^m,g)$ be complete, $m\geq3$, and suppose $u\gt{}0$ satisfies (5.2.6) with

\[ 0\leq\alpha\leq\frac{m-1}{m-2}. \]

\[ 0\lt{}u_-:=\inf_Mu\leq \sup_Mu=:u_+\lt{}\infty, \]

then

\begin{equation} \tag{5.2.18} \label{eq:AX-bounded-u-diam-from-geodesic} \mathrm{diam}(M) \leq \frac{\pi}{\sqrt\lambda} \left(\frac{u_+}{u_-}\right)^{\frac{m-3}{2(m-1)}\alpha}. \end{equation}

In particular $M$ is compact.

Proof. Because $u$ is bounded above and below, the weighted metric $u^{2\alpha}g$ is complete, so $L_u^\alpha$-minimizers exist. Let $\alpha:[0,l]\to M$ be an $L_u^\alpha$-minimizer from $p$ to $q$, parametrized by $g$-arclength, and let $x=\alpha(l/2)$. As above, the weighted excess has a local minimum at $x$.

Use (5.2.9) on each half of $\alpha$ with

\[ \psi(s)=u(x)^{\alpha/(m-1)}\sin\left(\frac{\pi s}{2(l/2)}\right). \]

Since $\alpha(m-3)/(m-1)\geq0$, we estimate the positive term by $u_+^{\frac{m-3}{m-1}\alpha}$ and the negative term by $u_-^{\frac{m-3}{m-1}\alpha}$. Thus

\[ 0\leq\Delta e(x) \leq (m-1)u(x)^{\frac{2\alpha}{m-1}} \left( \frac{u_+^{\frac{m-3}{m-1}\alpha}\pi^2}{2l} -\frac{\lambda u_-^{\frac{m-3}{m-1}\alpha} l}{2} \right). \]

Hence

\[ l\leq \frac{\pi}{\sqrt\lambda} \left(\frac{u_+}{u_-}\right)^{\frac{m-3}{2(m-1)}\alpha}. \]

Since $d_g(p,q)\leq l$ and $p,q$ were arbitrary, this proves (5.2.18). Compactness follows from Hopf–Rinow. ◻

Volume estimate.

Lemma 5.2.18

Assume the hypotheses of Theorem 5.2.13 and normalize $\min_{\widetilde N}\tilde u=1$. Set $\theta=2\alpha/(n-1)$ and define, on $\widetilde N$,

\begin{equation} \tag{5.2.19} \label{eq:AX-weighted-profile} I(v)=\inf\left\{ \int_{\partial E}\tilde u^\alpha: E\Subset\widetilde N,\quad \int_E \tilde u^\theta=v \right\}. \end{equation}

Then $I$ satisfies

\begin{equation} \tag{5.2.20} \label{eq:AX-profile-ODE-ineq} I''I\leq -\frac{(I')^2}{n-1}-(n-1)\lambda \end{equation}

in the viscosity sense.

Proof. By the diameter part, $\widetilde N$ is compact; hence the minimizer $E$ exists for every $v_0\in(0,\int_{\widetilde N}\tilde u^\theta)$. We suppress tildes. Put

\[ V(E)=\int_Eu^\theta,\qquad A(E)=\int_{\partial E}u^\alpha . \]

Let $\Sigma=\partial E$ and take a smooth normal variation with variational field $\varphi\nu$. We extend $\varphi$ to a neighborhood of $\Sigma$ and use the same symbol for the extension. With the convention used here,

\[ \partial_t d\mu_t=H\varphi\,d\mu,\qquad \partial_t\nu=-\nabla^\Sigma\varphi,\qquad \partial_t H=-\Delta^\Sigma\varphi -\bigl(|A_\Sigma|^2+\mathrm{Ric}_N(\nu,\nu)\bigr)\varphi . \]

Also

\[ \partial_t(u^{-1}u_\nu) = u^{-1}\mathrm{Hess}^N u(\nu,\nu)\varphi -u^{-1}\langle\nabla^\Sigma u,\nabla^\Sigma\varphi\rangle -u^{-2}u_\nu^2\varphi . \]

Thus

\[ V'(0)=\int_\Sigma u^\theta\varphi,\qquad V''(0)=\int_\Sigma (H+\theta u^{-1}u_\nu)u^\theta\varphi^2 +u^\theta\varphi\varphi_\nu, \]

\[ A'(0)=\int_\Sigma u^\alpha\varphi(H+\alpha u^{-1}u_\nu), \]

and differentiating this last expression gives the following formula. In the following displays, the same color marks the terms with the same origin: magenta terms cancel by integration by parts or by the normal derivative of $\varphi=u^{-\alpha}$; green terms are the spectral-Ricci pair; blue terms come from $|A_\Sigma|^2$; red terms come from $u_\nu^2$; orange terms come from $Hu_\nu$; and purple terms come from the surviving part of the last line.

\begin{equation} \tag{5.2.21} \begin{aligned} A''(0) ={}& \int_\Sigma \left( \begin{aligned} &\class{note-ax-cancel}{-\Delta^\Sigma\varphi} \class{note-ax-spec}{-\mathrm{Ric}_N(\nu,\nu)\varphi} \class{note-ax-geom}{-|A_\Sigma|^2\varphi} \end{aligned} \right) u^\alpha\varphi \notag\\ &+\int_\Sigma \left( \begin{aligned} &\class{note-ax-square}{-\alpha u^{-2}u_\nu^2\varphi} \class{note-ax-spec}{+\alpha u^{-1}\Delta^Nu\,\varphi} \class{note-ax-cancel}{-\alpha u^{-1}\Delta^\Sigma u\,\varphi}\\ &\class{note-ax-normal}{-\alpha u^{-1}Hu_\nu\varphi} \class{note-ax-cancel}{-\alpha u^{-1}\langle\nabla^\Sigma u,\nabla^\Sigma\varphi\rangle} \end{aligned} \right) u^\alpha\varphi \notag\\ &+\int_\Sigma \left( \begin{aligned} &\class{note-ax-cancel}{\alpha u^{\theta-1}u_\nu\varphi^2} \class{note-ax-cancel}{+u^\theta\varphi\varphi_\nu}\\ &\qquad \class{note-ax-last}{+Hu^\theta\varphi^2} \end{aligned} \right) u^{\alpha-\theta}\class{note-ax-last}{(H+\alpha u^{-1}u_\nu)}. \label{eq:AX-area-second-profile} \end{aligned} \end{equation}

The last line is just the derivative of the factor $u^\alpha\varphi\,d\mu_t$ in $A'(t)$, rewritten with a factor $u^\theta$:

\[ \partial_t(u^\alpha\varphi\,d\mu_t) = \left( \begin{aligned} &\class{note-ax-cancel}{\alpha u^{\alpha-1}u_\nu\varphi^2} \class{note-ax-cancel}{+u^\alpha\varphi\varphi_\nu}\\ &\qquad \class{note-ax-last}{+Hu^\alpha\varphi^2} \end{aligned} \right)d\mu . \]

The volume constraint implies that

\[ u^{\alpha-\theta}(H+\alpha u^{-1}u_\nu)=A_v'(v_0), \]

where $A_v$ is the area of this variation written as a function of $V$. Choose $\varphi=u^{-\alpha}$ and set

\[ Q=\int_\Sigma u^{\theta-\alpha},\qquad X=u^{\theta-\alpha}A_v'(v_0),\qquad Y=u^{-1}u_\nu . \]

Then $H=X-\alpha Y$. For $\varphi=u^{-\alpha}$, the tangential $\Delta^\Sigma u$ term cancels after integration by parts with the tangential gradient term

\[ \class{note-ax-cancel}{-\alpha u^{-1}\langle\nabla^\Sigma u,\nabla^\Sigma\varphi\rangle}. \]

Hence the first two lines of (5.2.21), together with (5.2.3) and $|A_\Sigma|^2\geq H^2/(n-1)$, are bounded above by

\[ \int_\Sigma u^{-\alpha} \left[ \class{note-ax-geom}{-\frac{H^2}{n-1}} \class{note-ax-normal}{-\alpha HY} \class{note-ax-square}{-\alpha Y^2} \class{note-ax-spec}{-(n-1)\lambda} \right]. \]

The last line of (5.2.21) is

\[ \int_\Sigma u^{-\alpha}\class{note-ax-last}{XH}, \]

because $\class{note-ax-cancel}{\varphi_\nu=-\alpha u^{-\alpha}Y}$ and $\class{note-ax-last}{H+\alpha Y=X}$. Therefore

\[ A''(0) \leq \int_\Sigma u^{-\alpha} \left[ \class{note-ax-geom}{-\frac{H^2}{n-1}} \class{note-ax-normal}{-\alpha HY} \class{note-ax-square}{-\alpha Y^2} \class{note-ax-spec}{-(n-1)\lambda} \class{note-ax-last}{+XH} \right]. \]

Expanding $H=X-\alpha Y$ gives

\[ \begin{aligned} A''(0) \leq{}& \int_\Sigma u^{-\alpha}\Big[ \class{note-ax-geom}{-\frac{X^2}{n-1}} \class{note-ax-geom}{+\frac{2\alpha}{n-1}XY} \class{note-ax-geom}{-\frac{\alpha^2}{n-1}Y^2} \class{note-ax-square}{-\alpha Y^2} \class{note-ax-normal}{-\alpha XY} \class{note-ax-normal}{+\alpha^2Y^2} \notag\\ &\qquad \class{note-ax-last}{+X^2} \class{note-ax-last}{-\alpha XY} \class{note-ax-spec}{-(n-1)\lambda} \Big]. \end{aligned} \]

The inverse-function chain rule gives

\[ A_v''(v_0) = (V'(0))^{-2}A''(0)-(V'(0))^{-3}A'(0)V''(0). \]

Substituting the formulas for $V',V''$ and using $\theta=2\alpha/(n-1)$ yields

\[ \begin{aligned} Q^2A_v''(v_0) \leq{}& \int_\Sigma u^{-\alpha}\left[ -\frac{X^2}{n-1} +\left(\frac{2\alpha}{n-1}-\theta\right)XY +\left(\frac{n-2}{n-1}\alpha^2-\alpha\right)Y^2 -(n-1)\lambda \right]\notag\\ \leq{}& -\left(\frac{A_v'(v_0)^2}{n-1}+(n-1)\lambda\right) \int_\Sigma u^{2\theta-3\alpha}. \end{aligned} \]

Here we used $\alpha\leq(n-1)/(n-2)$ and $u\geq1$. Holder’s inequality gives

\[ A_v(v_0)\int_\Sigma u^{2\theta-3\alpha} \geq \left(\int_\Sigma u^{\theta-\alpha}\right)^2=Q^2. \]

Thus

\[ A_v(v_0)A_v''(v_0) \leq -\frac{A_v'(v_0)^2}{n-1}-(n-1)\lambda . \]

Since $A_v$ is an upper barrier for $I$ at $v_0$, this is exactly the viscosity inequality (5.2.20). ◻

Lemma 5.2.19

Let $V\in(0,\infty]$ and let $I:[0,V)\to\mathbb R$ be continuous with $I(0)=0$ and $I(v)\gt{}0$ for $v\in(0,V)$. Suppose

\[ I''I\leq -\frac{(I')^2}{n-1}-(n-1)\lambda \]

in the viscosity sense on $(0,V)$, and suppose

\[ \limsup_{v\to0^+}v^{-\frac{n-1}{n}}I(v) \leq n|B^n|^{1/n}. \]

Then

\[ V\leq \lambda^{-n/2}|S^n|. \]

Proof. Set

\[ \psi=I^{\frac n{n-1}}. \]

Applying the chain rule to positive upper test functions for $I$, the preceding viscosity inequality is equivalent to

\begin{equation} \tag{5.2.22} \label{eq:AX-psi-ODE} \psi''\leq -n\lambda \psi^{\frac{2-n}{n}} \end{equation}

in the viscosity sense. The small-volume assumption gives

\[ \psi'_+(0):=\limsup_{v\to0^+}\frac{\psi(v)}{v} \leq n^{\frac n{n-1}}|B^n|^{\frac1{n-1}} =n|S^{n-1}|^{\frac1{n-1}}. \]

For $\zeta\gt{}0$ define

\[ \mu(r)=\frac{\sin(\sqrt\lambda r)}{\sqrt\lambda}, \qquad v_\zeta(r)=\zeta\int_0^r\mu(s)^{n-1}\,ds, \qquad 0\leq r\leq\frac{\pi}{\sqrt\lambda}, \]

and define the model profile $I_\zeta$ by

\[ I_\zeta(v_\zeta(r))=\zeta\mu(r)^{n-1}. \]

Let $\psi_\zeta=I_\zeta^{n/(n-1)}$. Since

\[ \psi_\zeta(v_\zeta(r))=\zeta^{\frac n{n-1}}\mu(r)^n, \qquad \frac{dv_\zeta}{dr}=\zeta\mu(r)^{n-1}, \]

direct differentiation gives

\[ \psi_\zeta''=-n\lambda\psi_\zeta^{\frac{2-n}{n}}, \qquad (\psi_\zeta)'_+(0)=n\zeta^{\frac1{n-1}}. \]

The existence interval of $I_\zeta$ has length

\[ V_\zeta = \zeta\int_0^{\pi/\sqrt\lambda}\mu(s)^{n-1}\,ds. \]

For $\zeta=|S^{n-1}|$, this is exactly $\lambda^{-n/2}|S^n|$.

Assume by contradiction that $V\gt{}\lambda^{-n/2}|S^n|$. Choose $\zeta\gt{}|S^{n-1}|$ so close to $|S^{n-1}|$ that $V_\zeta\lt{}V$. Then $(\psi_\zeta)'_+(0)\gt{}\psi'_+(0)$, so $\psi\lt{}\psi_\zeta$ on $(0,\delta)$ for some $\delta\gt{}0$. If the two functions first meet at $a\in(0,V_\zeta)$, then on $(0,a)$ we have $\psi\lt{}\psi_\zeta$. Since the function $s\mapsto -n\lambda s^{(2-n)/n}$ is increasing on $(0,\infty)$, the difference $w=\psi-\psi_\zeta$ satisfies $w''\lt{}0$ in the viscosity sense on $(0,a)$. Thus $w$ is strictly concave there. But $w(0)=w(a)=0$ and $w\lt{}0$ on $(0,a)$, which is impossible for a concave function. Hence $\psi\lt{}\psi_\zeta$ on $(0,V_\zeta)$. Since $\psi_\zeta(V_\zeta)=0$ while $V_\zeta\lt{}V$ and $I\gt{}0$ on $(0,V)$, continuity gives a contradiction at $V_\zeta$. Therefore $V\leq \lambda^{-n/2}|S^n|$. ◻

Proof. Normalize $\min_{\widetilde N}\tilde u=1$ and set

\[ V_0=\int_{\widetilde N}\tilde u^\theta . \]

The profile $I$ satisfies (5.2.20). Since $\tilde u$ attains its minimum at some point $\tilde p$, small geodesic balls centered at $\tilde p$ satisfy

\[ \int_{B_r(\tilde p)}\tilde u^\theta=|B^n|r^n+O(r^{n+1}), \qquad \int_{\partial B_r(\tilde p)}\tilde u^\alpha=n|B^n|r^{n-1}+O(r^n). \]

Since these balls are admissible competitors for $I$, it follows that

\[ \limsup_{v\to0}v^{-\frac{n-1}{n}}I(v)\leq n|B^n|^{1/n}. \]

Lemma 5.2.19 applied with $V=V_0$ therefore gives

\[ V_0\leq \lambda^{-n/2}|S^n|. \]

Since $\tilde u\geq1$, we have

\[ |\widetilde N|\leq\int_{\widetilde N}\tilde u^\theta=V_0 \leq \lambda^{-n/2}|S^n|. \]

If equality holds, then $\int_{\widetilde N}\tilde u^\theta=|\widetilde N|$ and $\tilde u\geq1$, hence $\tilde u\equiv1$. The spectral inequality becomes the pointwise Ricci bound $\mathrm{Ric}_{\widetilde N}\geq(n-1)\lambda$, and the equality case in the classical Bishop–Gromov theorem gives that $\widetilde N$ is the round sphere of radius $\lambda^{-1/2}$. ◻

We need a comparison of distance.

Lemma 5.2.20

Let $\varphi:M\to \mathbb{R}^{n+1}$ be the immersion and $\tilde{g}=r^{-2}g$ and $g=\varphi^*(\delta)$ with $0 \in M$. Given two points $p,q\in M$ with $d_{\tilde{g}}(p,q)\leq D$, we have $r(p)\leq e^{D}r(q)$.

Proof. Let $\alpha$ be a curve joining $p$ and $q$ with length $D+\varepsilon$. Then, we have

\[ \begin{aligned} \log r(p)-\log r(q)={} & \int_{ } \frac{\left< \nabla r, \alpha'(t) \right> }{r(\alpha(t))}dt\leq \int_{ } \frac{|\alpha'(t)|_g}{r}dt \\ ={} & \int_{ } |\alpha'(t)|_{\tilde{g}}d\tilde{t}=D+\varepsilon. \end{aligned} \]

◻

Proof. Let $M^n\hookrightarrow\mathbb{R}^{n+1}$ be as in the theorem, where $n=3$ or $n=4$. We fix any point $x_0\in M$ and suppose $x_0=0$ after a translation. Let $\tilde{g}=r^{-2}g$, where $r(x)=|x|$.

The preceding spectral estimates and $\mu$-bubble constructions give the following dimension-dependent constants. There exist $L_n,A_n,D_n\lt{}\infty$ such that, whenever $N$ is a compact collar in $(M,\tilde{g})$ with $d_{\tilde{g}}(\partial_-N,\partial_+N)\geq L_n$, one can find a smooth $\mu$-bubble component $\Sigma$ which separates $\partial_-N$ from $\partial_+N$, lies in $\tilde B_{L_n}(\partial_-N)$, and satisfies

\[ |\Sigma|_{\tilde{g}}\leq A_n,\qquad \mathrm{diam}_{\tilde{g}}(\Sigma)\leq D_n . \]

For $n=3$ and $n=4$ this follows from the $\mu$-bubble reduction in Theorem 5.2.8 together with the spectral Ricci estimates in §5.2.6.

For any $\rho\gt{}0$, choose $R$ large such that

\[ d_{\tilde{g}}(\partial B_\rho^M,\partial B_R^M)\geq L_n. \]

Set $N=B_R^M\setminus B_\rho^M$, with $\partial_-N=\partial B_\rho^M$ and $\partial_+N=\partial B_R^M$, and apply the preceding paragraph. If $x\in\Sigma$, then there is $y\in\partial B_\rho^M$ with $d_{\tilde{g}}(x,y)\leq L_n$. By the distance comparison lemma,

\[ r(x)\leq e^{L_n}r(y)\leq e^{L_n}\rho . \]

Therefore $\Sigma\subset B_{C_n\rho}$ in the Euclidean metric. Since $g=r^2\tilde{g}$, the induced measures on the $(n-1)$-dimensional hypersurface $\Sigma$ satisfy $d\mu_g=r^{n-1}d\mu_{\tilde{g}}$, and hence

\[ |\Sigma|_g \leq (C_n\rho)^{n-1}|\Sigma|_{\tilde{g}} \leq C_n\rho^{n-1}. \]

Since $M$ is simply connected and has one end [CSZ97], the side of $\Sigma$ containing $B_\rho^M(x_0)$ is a compact region. Let $\Omega_\rho$ denote this region; then $B_\rho^M(x_0)\subset\Omega_\rho$ and $\partial\Omega_\rho=\Sigma$. The Michael–Simon isoperimetric inequality for minimal submanifolds [MS73] gives

\[ |\Omega_\rho|_g\leq C_n|\Sigma|_g^{\frac{n}{n-1}}. \]

Consequently,

\[ |B_\rho^M(x_0)| \leq |\Omega_\rho|_g \leq C_n|\Sigma|_g^{\frac{n}{n-1}} \leq C_n\rho^n. \]

◻

Mazet’s proof in the ambient space $\mathbb{R}^6$

Here $\mathbb{R}^6$ means the hypersurface dimension is $5$. Mazet [Maz24] proves that every complete, connected, two-sided stable minimal immersion

\[ M^5\hookrightarrow \mathbb{R}^6 \]

is flat. The proof follows the Chodosh–Li–Minter–Stryker strategy [CLMS25], but with one extra parameter in the curvature quantity. This extra parameter is the weighted bi-Ricci curvature.

Let $(N^m,g)$ be a Riemannian manifold, and let $\{e_1,\ldots,e_m\}$ be an orthonormal basis. For $\alpha\in\mathbb{R}$, Mazet defines the $\alpha$-weighted bi-Ricci curvature by

\[ \operatorname{BRic}_{\alpha}(e_1,e_2) := \sum_{i=2}^{m}R(e_1,e_i,e_i,e_1) + \alpha \sum_{j=3}^{m}R(e_2,e_j,e_j,e_2). \]

Its pointwise minimum is

\[ \Lambda_\alpha(p) := \min_{\substack{e,f\in T_pN\\ |e|=|f|=1,\ \langle e,f\rangle=0}} \operatorname{BRic}_{\alpha}(e,f). \]

When $\alpha=1$, this is the usual bi-Ricci curvature:

\[ \operatorname{BRic}_{1}(e_1,e_2) = \operatorname{Ric}(e_1,e_1) + \operatorname{Ric}(e_2,e_2) - R(e_1,e_2,e_2,e_1). \]

Thus $\alpha$ lets one change the relative weight of the curvature directions which are tangent to the eventual $\mu$-bubble.

The first step is the same conformal change as before. Remove the points where the immersion hits the origin and set

\[ \tilde g=r^{-2}g, \qquad N:=M\setminus F^{-1}(0). \]

The Gulliver–Lawson observation is that $(N,\tilde g)$ is complete. Mazet’s main spectral estimate says that, in dimension $5$, one can choose

\[ a=\frac{11}{10}, \qquad \alpha=\frac{40}{43}, \qquad \delta=\frac{3}{10}, \]

so that the stability inequality implies the following weighted bi-Ricci spectral lower bound:

\[ \int_N |\tilde\nabla\varphi|^2\,d\tilde\mu \geq \frac1a\int_N V\varphi^2\,d\tilde\mu, \qquad V\geq \delta-\widetilde{\Lambda}_\alpha, \qquad \varphi\in C_c^1(N). \]

Equivalently,

\[ \int_N \left( a|\tilde\nabla\varphi|^2 +(\widetilde{\Lambda}_\alpha-\delta)\varphi^2 \right)d\tilde\mu \geq0. \]

So $(N,\tilde g)$ does not have a pointwise lower bound $\widetilde{\Lambda}_\alpha\geq\delta$, but it has the corresponding spectral lower bound.

The second step is to build a weighted $\mu$-bubble in a long annulus of $(N,\tilde g)$. The bubble is a hypersurface

\[ \Sigma^4=\partial\Omega \]

which separates the inner and outer boundary of the annulus. The second variation of the weighted $\mu$-bubble turns the spectral $\operatorname{BRic}_\alpha$ bound on $N$ into a spectral Ricci bound on $\Sigma$. More precisely, if

\[ \lambda_\Sigma(x):=\min_{|v|=1}\operatorname{Ric}_\Sigma(v,v), \]

then the induced metric on $\Sigma$ satisfies an inequality of the form

\[ \int_\Sigma \left( \frac{4}{(4-a)\alpha}|\nabla\phi|^2 + \lambda_\Sigma\phi^2 \right)d\mu_\Sigma \geq \frac{\delta}{2\alpha}\int_\Sigma \phi^2\,d\mu_\Sigma. \]

For Mazet’s parameters,

\[ \frac{4}{(4-a)\alpha}=\frac{43}{29}\lt{}\frac32 = \frac{k-1}{k-2}, \qquad k=\dim\Sigma=4. \]

This is the numerical point that allows the Antonelli–Xu spectral Bishop–Gromov theorem [AX24] to be applied to the $4$-dimensional bubble. It gives a uniform $\tilde g$-volume bound for $\Sigma$.

Finally one returns to the original metric $g$. If $\Sigma$ is chosen around $B_\rho^M(p_0)$, the conformal collar bound gives

\[ r\leq C\rho \qquad\text{on }\Sigma, \]

so the $\tilde g$-volume bound for $\Sigma$ becomes a $g$-area bound

\[ |\Sigma|_g\leq C\rho^4. \]

Since $M$ has one end, one can choose the relevant component of $\Sigma$ to enclose $B_\rho^M(p_0)$. The Michael–Simon–Brendle isoperimetric inequality [MS73, Bre21] on minimal hypersurfaces then gives

\[ |B_\rho^M(p_0)|\leq C|\Sigma|_g^{5/4}\leq C\rho^5. \]

This is the Euclidean volume growth needed by the Schoen–Simon–Yau stable Bernstein theorem. Hence $M^5$ is flat. In short, the new feature in $\mathbb{R}^6$ is that the weighted curvature $\operatorname{BRic}_{40/43}$ creates just enough spectral Ricci positivity on the $4$-dimensional $\mu$-bubble for Antonelli–Xu’s volume estimate to close.

Key Steps in the Regularity Proofs

Mon, 29 Jun 2026 00:00:00 +0000

Up to a subsequence, we can assume $|M_k|$ or $V_k$ converges to $V$ in the varifold sense. Pick any point $x \in \mathrm{spt}\|V\|$. We need to analyze the tangent cone of $V$ at $x$. There are several cases:

Hyperplanes. We need to develop a sheeting theorem to show the regularity and smooth convergence.
Classical Cones. We need a minimum distance theorem (embedded case) or a decomposition theorem (immersed case) to show the regularity and smooth convergence.
$\boldsymbol{C}\times \mathbb{R}^m$ where $\boldsymbol{C}$ is a stable cone with isolated singularity. Classification of stable cones.
other cones. We need dimension reduction argument to reduce to the previous case.

Sets of Finite Perimeter

Mon, 29 Jun 2026 00:00:00 +0000

We collect some basic definitions and facts about the set of finite perimeter.

If $E$ is a set with smooth boundary, then we have the following Gauss-Green formula:

\[ \int_{E} \mathrm{div} \varphi d\mathcal{H}^n=\int_{ \partial E} \varphi \cdot \nu d\mathcal{H}^{n-1}, \]

for any $\varphi \in C_c^1(\mathbb{R}^n,\mathbb{R}^n)$. In particular, if we require that $|\varphi|\leq 1$, then the right-hand side can be bounded by the perimeter of $E$ as

\[ \int_{ \partial E} \varphi\cdot \nu d\mathcal{H}^{n-1}\leq \mathcal{H}^{n-1}(\partial E). \]

and equality holds if and only if $\varphi=\nu$ on $\partial E$. This fact motivates the following definition of the perimeter of $E$ as

\[ |\partial E|= \sup_{\varphi \in C_c^1(\mathbb{R}^n,\mathbb{R}^n), |\varphi|\leq 1} \int_{ \partial E} \varphi\cdot \nu d\mathcal{H}^{n-1}. \]

Note that the right-hand side of the equation above does not depend on the regularity of the boundary of $E$. This motivates the following definition.

Definition 3.2.1

The perimeter of a set $E$ in an open set $U$ is defined as

\[ P(E,U)= \sup_{\varphi \in C_c^1(U,\mathbb{R}^n), |\varphi|\leq 1} \int_{ U\cap E} \mathrm{div} \varphi d\mathcal{H}^n. \]

We say that $E$ has locally finite perimeter in $U$ if $P(E,W)\lt{}+\infty$ for any $W \subset \subset U$. Such a set is also called a Caccioppoli set.

Suppose $E$ has locally finite perimeter in $U$. Then, we can consider the linear functional $J_E$ on $C_c^1(U,\mathbb{R}^n)$ defined by

\[ J_E(\varphi)= \int_{ U\cap E} \mathrm{div} \varphi d\mathcal{H}^n. \]

This functional is clearly linear. Since we have assumed that $E$ has locally finite perimeter, we have that $J_E$ is a bounded linear functional on $C_c^1(W,\mathbb{R}^n)$ for any $W \subset \subset U$. (Recall that an operator $T$ is bounded if $\sup_{|\varphi|\leq 1} |T(\varphi)|\lt{}+\infty$.) Now, we can apply the Riesz Representation Theorem to get a unique Radon measure $\mu_E$ on $U$, and a vector-valued function $\nu_E$ on $U$ such that $|\nu_E(x)|=1$ for $\mu_E$-almost every $x$ and

\[ J_E(\varphi)= \int_{ U} \varphi\cdot \nu_E d\mu_E, \]

for any $\varphi \in C_c^1(U,\mathbb{R}^n)$.

The vector-valued measure $\nu_E \mu_E$ is called the Gauss-Green measure of $E$ in $U$. We use $\overrightarrow{\mu}_E$ to denote the vector-valued measure $\nu_E \mu_E$. Then, the perimeter of $E$ in $U$ can also be written as

\[ P(E,U)= \mu_E(U). \]

We can understand $\mu_E$ as a boundary measure of $E$ in $U$, and $\nu_E$ as a boundary normal vector of $E$ in $U$, pointing outward.

Suppose $E=[0,+\infty)^{2} \in \mathbb{R}^2$. Then, $E$ is a set with locally finite perimeter. In particular,

\[ \mu_E=\mathcal{H}^1|_{[0,+\infty)\times \{0\}} + \mathcal{H}^1|_{\{0\}\times [0,+\infty)},\quad \nu_E=(0,-1)|_{[0,+\infty)\times \{0\}} + (1,0)|_{\{0\}\times [0,+\infty)}. \]

Recall that $\mathrm{spt}\,\mu_E$ is the support of $\mu_E$, which is defined as the set of points $x$ such that $\mu_E(B_r(x))\gt{}0$ for any $r\gt{}0$.

Then, we have the following proposition:

Proposition 3.2.2

Suppose $E$ is a set of locally finite perimeter in $\mathbb{R}^n$. Then, $\mathrm{spt}\,\mu_E\subset \partial E$, where $\partial E$ is the topological boundary of $E$.

Let

\[ E := (0,1)^2 \backslash \{ 0.5 \}\times [0,0.5]\subset \mathbb{R}^2. \]

Then $\partial E$ contains both the outer boundary of the square and the slit:

\[ \partial E=\partial(0,1)^2\;\cup\;\bigl(\{0.5\}\times[0,0.5]\bigr). \]

On the other hand, removing a $1$-dimensional set does not change $\chi_E$ in $L^1$, so the perimeter measure is the same as for the open square:

\[ \mu_E=\mu_{(0,1)^2},\qquad \mathrm{spt}\,\mu_E=\partial(0,1)^2. \]

Hence

\[ \mathrm{spt}\,\mu_E \subsetneq \partial E. \]

To better illustrate the “true” boundary of a set of finite perimeter in the measure sense, we introduce the reduced boundary.

Definition 3.2.3

Given a set $E$ of locally finite perimeter in $\mathbb{R}^n$, the reduced boundary $\partial^*E$ is the set of points $x\in\mathbb{R}^n$ such that the limit

\[ \nu_E(x):=\lim_{r \to 0^+} \frac{\overrightarrow{\mu}_E(B_r(x))}{\mu_E(B_r(x))} \]

exists and belongs to $\mathbb{S}^{n-1}$.

For the $E=(0,1)^2 \backslash \{ 0.5 \}\times [0,0.5]$, we have

\[ \partial^* E = \partial(0,1)^2\backslash \{ (0,0), (1,0), (0,1), (1,1) \}. \]

In particular, we have

\[ \lim_{r \to 0^+} \frac{\overrightarrow{\mu}_E(B_r(0))}{\mu_E(B_r(0))} = (-\frac{1}{2},-\frac{1}{2})\notin \mathbb{S}^{n-1}. \]

So $(0,0)\notin \partial^* E$.

We have the following structure theorem for the reduced boundary.

Theorem 3.2.4

Suppose $E$ is a set of locally finite perimeter in $\mathbb{R}^n$. Then, the reduced boundary $\partial^* E$ is a $(n-1)$-countably rectifiable set in $\mathbb{R}^n$, and we actually have

\[ \mu_E = \mathcal{H}^{n-1}|_{\partial^* E},\quad \nu_E = \text{ the outer normal vector field of } \partial^* E. \]

Note that since $\partial^* E$ is rectifiable, so the approximate tangent space $T_x \partial^* E$ is well-defined for $\mu_E$-almost every $x \in \partial^* E$. Hence, we can choose the outer normal vector field perpendicular to $T_x \partial^* E$ and point outward of $E$.

Definition 3.2.5

Given Lebesgue measurable sets $\{E_h\}_{h\in\mathbb{N}}$ and $E$ in $\mathbb{R}^n$, we say that $E_h$ locally converges to $E$, and write $E_h \xrightarrow{\mathrm{loc}} E$, if

\[ \lim_{h \to \infty} \left| K \cap (E \Delta E_h) \right| = 0, \qquad \forall K \subset \mathbb{R}^n \text{ compact.} \]

This is equivalent to say that $\chi_{E_h} \xrightarrow{} \chi_{E}$ in $L^1_{loc}(\mathbb{R}^n)$.

Proposition 3.2.6

If $\{E_h\}_{h\in\mathbb{N}}$ is a sequence of sets of locally finite perimeter in $\mathbb{R}^n$, with

\[ E_h \xrightarrow{\mathrm{loc}} E, \qquad \limsup_{h \to \infty} P(E_h; K) \lt{} \infty, \]

for every compact set $K$ in $\mathbb{R}^n$, then $E$ is of locally finite perimeter in $\mathbb{R}^n$, $\mu_{E_h} \rightharpoonup^* \mu_E$ and, for every open set $A \subset \mathbb{R}^n$, we have

\[ P(E;A) \leq \liminf_{h \to \infty} P(E_h;A). \]

The inequality above can be strict.

Let $B$ denote the closed unit disc $B = \{x \in \mathbb{R}^2 : |x| \leq 1\}$. For each $i \in \mathbb{N}$, let $E_i = B \setminus \bigcup_{k=1}^{n_i} \bar{B}_{r_i}(x_{i,k})$, where $\{x_{i,k}\}_{k=1}^{n_i}$ is a collection of centers inside $B$, $n_i$ increases as $i \to \infty$, and $r_i\to 0$ in such a way that $n_i r_i\to c\gt{}0$ and

\[ \bigcup_{k=1}^{n_i} \bar{B}_{r_i}(x_{i,k}) \subset B, \]

with all the small discs disjoint and contained in $B$.

For each $E_i$, the perimeter is

\[ P(E_i) = P(B) + n_i P(\bar{B}_{r_i}) = 2\pi + n_i \cdot 2\pi r_i, \]

since each removed disc adds $2\pi r_i$ to the perimeter.

As $i \to \infty$, the number of holes $n_i \to \infty$ and the radii $r_i \to 0$, so the total removed area goes to $0$. In the limit, the set $E_i$ converges (locally in measure, or in $L^1_{\mathrm{loc}}$) to $B$. However,

\[ \lim_{i \to \infty} P(E_i) = 2\pi+2\pi c, \]

because the total boundary length of the small holes remains visible before passing to the limit.

Thus, the perimeter functional is lower semicontinuous under local convergence, and it is possible that

\[ P(\lim_{i\to\infty} E_i) \lt{} \liminf_{i\to\infty} P(E_i), \]

because the limiting set does not retain the interior boundaries present in the approximating sequence.

Let $D = [0,4] \times [0,2] \subset \mathbb{R}^2$. For each $i\in\mathbb{N}$, construct $E_i$ as the subset of $D$ with lower and side boundaries of $D$, but the top replaced with a zig-zag curve of $n_i$ “teeth,” each of amplitude $h_i$ (height above $y=2$), so that as $i\to\infty$, $n_i\to\infty$ and $h_i\to 0$.

Each $E_i$ is the set (shaded in the pictures above) contained below its zig-zag boundary and above $y=0$. As $i\to\infty$, the upper boundary of $E_i$ converges to the straight line $y=2$, and $E_i$ converges (in measure) to $D$.

The perimeter of $E_i$ is

\[ P(E_i) = 2\cdot 2 + 4 + L_{\text{zigzag}}, \]

where $L_{\text{zigzag}}$ is the total length of the zig-zag curve. If each tooth projects horizontally $\delta x = 4/n_i$ and has vertical height $h_i$, then

\[ L_{\text{zigzag}} = n_i \cdot 2\sqrt{ \left(\frac{\delta x}{2}\right)^2 + h_i^2 } = n_i \cdot 2 \sqrt{ (2/n_i)^2 + h_i^2 }. \]

If $n_i h_i\to a\gt{}0$, then the zig-zag boundary length satisfies

\[ \lim_{i\to\infty} L_{\text{zigzag}} = 2\sqrt{4+a^2}\gt{}4. \]

Thus, in the limit, the region $E_i$ converges to $D$, but the perimeter keeps a positive excess:

\[ P(\lim_{i\to\infty} E_i) \lt{} \liminf_{i\to\infty} P(E_i). \]

This example again shows lower semicontinuity: the limiting domain loses the additional oscillating boundary length in the limit.

Theorem 3.2.7

If $R\gt{}0$ and $(E_k)_{k\in\mathbb{N}}$ are sets of finite perimeter in $\mathbb{R}^n$, with

\[ \begin{aligned} \sup_{k\in\mathbb{N}} P(E_k) &\lt{} \infty,\\ E_k &\subset B_R,\qquad \forall\, k\in\mathbb{N}, \end{aligned} \]

then one can find a set $E$ of finite perimeter in $\mathbb{R}^n$ and a subsequence of $(E_k)_{k\in\mathbb{N}}$, still denoted by $E_k$, such that

\[ \chi_{E_{k}} \to \chi_E,\qquad \mu_{E_{k}} \stackrel{*}{\rightharpoonup} \mu_E,\qquad E \subset B_R. \]

Definition 3.2.8

Suppose $A\subset \mathbb{R}^n$ is a bounded set and $E_0$ is a set of finite perimeter in $\mathbb{R}^n$. We say $E_0$ is a perimeter minimizer in $A$ if

\[ P(E_0;A)\leq P(E;A) \]

for every set of finite perimeter $E$ such that $E\setminus A=E_0\setminus A$.

In particular, we say $E_0$ is a perimeter minimizer in $\mathbb{R}^n$ if $E_0$ is a perimeter minimizer for any bounded set $A$.

Proposition 3.2.9

Suppose $\{ E_k \}$ is a sequence of perimeter minimizers in $\mathbb{R}^n$, and assume $E_k$ converges to $E$ in $L^1_{\mathrm{loc}}(\mathbb{R}^n)$. Then $E$ is also a perimeter minimizer in $\mathbb{R}^n$.

Now, we are ready to state the general existence of minimizers in the following sense.

Proposition 3.2.10

Let $A \subset \mathbb{R}^n$ be a bounded set and let $E_0$ be a set of finite perimeter in $\mathbb{R}^n$. Then there exists a set of finite perimeter $E$ such that $E \setminus A = E_0 \setminus A$ and

\[ P(E) \leq P(F) \]

for every $F$ such that $F \setminus A = E_0 \setminus A$.

For example, given a boundary curve $\Gamma$, which lies on a boundary of a convex domain $D$. We can construct a set of finite perimeter $E$ such that $\partial(\partial E \cap \Omega) = \Gamma$ and it is the one with the minimal perimeter.

The key here is actually the regularity of the boundary of the set of finite perimeter.

Submanifolds

Mon, 29 Jun 2026 00:00:00 +0000

Definition 1.2.1

Let $(M^N, g)$ be a Riemannian manifold and $\iota: \Sigma^n \hookrightarrow M$ be a smooth immersion. The induced metric on $\Sigma$ is defined by

\[ \bar{g} = \iota^* g, \]

so that $\bar{g}(X,Y) = g(d\iota(X), d\iota(Y))$ for tangent vectors $X, Y \in T\Sigma$.

Definition 1.2.2

The vector-valued second fundamental form $\vec{A}$ is defined by

\[ \vec{A}(X,Y) = (\nabla_X Y)^\perp, \]

where $\nabla$ is the Levi-Civita connection of $M$ and $(\cdot)^\perp$ denotes the projection onto the normal bundle of $\Sigma$ in $M$. The mean curvature vector $\vec{H}$ is defined as the trace of $\vec{A}$:

\[ \vec{H}=\mathrm{Tr}(\vec{A}) = \sum_{i=1}^{n} \vec{A}(e_i, e_i), \]

where $\{e_i\}$ is an orthonormal basis of $T\Sigma$ with respect to the induced metric $\bar{g}$.

Definition 1.2.3

A submanifold $\Sigma$ is called a hypersurface when $\mathrm{codim}(\Sigma) = 1$, i.e., $n = N-1$.

If we assume $\Sigma$ is a two-sided hypersurface, then there exists a globally defined unit normal vector field $\nu$ on $\Sigma$. In this case, the second fundamental form can be expressed as a scalar-valued symmetric $(0,2)$-tensor $A$ defined by

\[ A(X,Y) = g(\vec{A}(X,Y), \nu) = -g(\nabla_X \nu, Y) = g(\nabla_X Y, \nu). \]

Then the mean curvature $H=g(\vec{H}, \nu)$ is the trace of $A$.

The eigenvalues $\kappa_1, \ldots, \kappa_{n}$ of $A$ are called the principal curvatures.

Gauss equation for hypersurfaces.

For a hypersurface $\Sigma$ in a Riemannian manifold $(M^N, g)$, the Gauss equation relates the intrinsic curvature of $\Sigma$ to the extrinsic curvature and the ambient curvature:

\[ R_\Sigma(X,Y,Z,W) = R_M(X,Y,Z,W) + A(X,Z)A(Y,W) - A(X,W)A(Y,Z), \]

3. From the geometric inequality to a numerical criterion

Thu, 27 Aug 2026 00:00:00 +0000

We selected orthonormal Euclidean directions $e_1,\ldots,e_\ell$, set

\[ x=(\nu_1,\ldots,\nu_\ell)^{\mathsf T}, \qquad \nu_i=\ip{\nu}{e_i}, \]

and began with an arbitrary positive function

\[ \tag{3.1}\label{eq:direct-G-model} g^2=G(\nu_1,\ldots,\nu_\ell)=G(x). \]

The aim was to check directly whether

\[ \abs{A}^2g^2+g\Delta_M g\geq c\abs{A}^2 \]

could hold with $c\gt{}0$.

Since $M$ is minimal, its Gauss-map components satisfy

\[ \Delta_M\nu_i=-\abs{A}^2\nu_i, \qquad 1\leq i\leq\ell. \]

Write

\[ G_i=\frac{\partial G}{\partial\nu_i}, \qquad G_{ij}=\frac{\partial^2G}{\partial\nu_i\partial\nu_j}. \]

Applying the chain rule to (3.1) gives

\[ \tag{3.2}\label{eq:direct-coordinate-identity} \class{key-formula-math}{\abs{A}^2g^2+g\Delta_M g =W\abs{A}^2- \sum_{i,j=1}^\ell\alpha_{ij}\ip{\nabla\nu_i}{\nabla\nu_j},} \]

where

\[ \tag{3.3}\label{eq:direct-W} W=G-\frac{1}{2}\sum_{i=1}^\ell\nu_iG_i, \qquad \alpha_{ij}=-\frac{1}{2}G_{ij}+\frac{G_iG_j}{4G}. \]

The remaining task was to find a constant $s$ such that

\[ \class{key-formula-math}{\sum_{i,j=1}^\ell\alpha_{ij}\ip{\nabla\nu_i}{\nabla\nu_j} \leq s\abs{A}^2.} \]

The remaining term in (3.2) was organized in a frame adapted to the selected directions. Put

\[ e_i^\top=e_i-\nu_i\nu, \qquad 1\leq i\leq\ell, \]

and let

\[ \pi=\operatorname{span}\{e_1^\top,\ldots,e_\ell^\top\} \subset T_pM. \]

At an interior point $\abs{x}\lt{}1$, the vectors $e_1^\top,\ldots,e_\ell^\top$ are linearly independent. To keep this exploratory reduction as simple as possible, we imposed the additional ansatz that $\pi$ is spanned by principal-curvature directions. We may then choose an orthonormal principal frame $\tau_1,\ldots,\tau_n$ such that $\pi=\operatorname{span}\{\tau_1,\ldots,\tau_\ell\}$, and write

\[ \tag{3.4}\label{eq:active-principal-frame} A(\tau_a,\tau_b)=\lambda_a\delta_{ab}, \qquad 1\leq a,b\leq n. \]

Equation (3.4) gives

\[ \tag{3.5}\label{eq:gradient-nu-principal-frame} \nabla\nu_i =-\sum_{a=1}^\ell \lambda_a\ip{e_i}{\tau_a}\tau_a, \qquad \sum_{a=1}^\ell \ip{e_i}{\tau_a}\ip{e_j}{\tau_a} =\delta_{ij}-\nu_i\nu_j. \]

Consequently,

\[ \sum_{i,j=1}^\ell\alpha_{ij}\ip{\nabla\nu_i}{\nabla\nu_j} =\sum_{a=1}^\ell B_{aa}\lambda_a^2, \]

where the coefficients required by the numerical calculation are given directly by

\[ \tag{3.6}\label{eq:B-entrywise} B_{aa} =-\frac{1}{2}\sum_{i,j=1}^\ell G_{ij}\ip{e_i}{\tau_a}\ip{e_j}{\tau_a} +\frac{1}{4G} \left(\sum_{i=1}^\ell G_i\ip{e_i}{\tau_a}\right)^2. \]

For simplicity, assume $n\gt{}\ell$ and $\lambda_{\ell+1}=\cdots=\lambda_n$. By minimality, the sharp choice of $s$ is the smallest number such that

\[ \sum_{a=1}^\ell B_{aa}\lambda_a^2 \leq s\left( \sum_{a=1}^\ell\lambda_a^2 +\frac{(\lambda_1+\cdots+\lambda_\ell)^2}{n-\ell} \right). \]

Define

\[ \tag{3.7}\label{eq:threshold-polynomials} p(t)=\prod_{a=1}^\ell(t-B_{aa}), \qquad P(t)=(n-\ell)p(t)+tp'(t). \]

The sharp value of $s$ is the largest real root of $P$; thus we take

\[ \tag{3.8}\label{eq:s-largest-root} s=\max\{t\in\R\colon P(t)=0\}. \]

Hence

\[ \abs{A}^2g^2+g\Delta_M g\geq (W-s)\abs{A}^2. \]

Numerical verification routine

A numerical verification routine takes

\[ x\in\left\{\abs{x}\leq 1\colon G(x)\gt{}0\right\} \]

and samples the coefficients $\ip{e_i}{\tau_a}$ subject to the second identity in (3.5). It evaluates $G,G_i,G_{ij}$, constructs $W,\alpha,B_{aa},p,P,s$ by (3.3)–(3.8), and computes

\[ \mathfrak m_{n,\ell}=W-s. \]

The candidate passes the sampled test precisely when the sampled minimum of $\mathfrak m_{n,\ell}$ is positive.

Warning 3.1

The numerical experiments were used only to identify which test functions were promising. Even when a candidate passed every sampled test, the required uniform inequality still had to be verified by rigorous mathematical calculation.

Dimension Reduction and Regularity of Perimeter Minimizers

Mon, 29 Jun 2026 00:00:00 +0000

Theorem 3.3.1

Suppose $E$ is a perimeter minimizer in $\mathbb{R}^{n+1}$. We denote $V=|\partial^* E|$. Then, we have the following regularity result: $\mathrm{sing}\|V\|=\emptyset$ if $n\leq 6$, $\mathrm{sing}\|V\|$ is discrete if $n=7$, and $\mathcal{H}^{n-7+\delta}(\mathrm{sing}\|V\|)=0$ for any $\delta\gt{}0$ for $n\geq 8$.

Theorem 3.3.2

Let $\mathcal{V}$ be the collection of all the varifolds in the preceding theorem, i.e., the varifolds corresponding to the perimeter minimizers. Then,

\[ \operatorname{dim}(\mathrm{sing}(\|V\|\cap B_1))\leq n-7. \]

In particular, if $n=7$, then $\mathrm{sing}(\|V\|\cap B_1)$ is discrete.

Proof. We denote $F^l=\{ V \in \mathcal{V}: \mathcal{H}^l(\mathrm{sing}\cap B_1)\gt{}0 \}$.

Proposition 3.3.3

For each $V \in F^l$, there exists $\boldsymbol{C} \in \mathrm{VarTan}(V,x)\cap F^l$ for $\mathcal{H}^l-$a.e. $x \in \mathrm{sing}(\|V\|)\cap B_1$.

Proof. Recall that we actually have for $\mathcal{H}^l$-a.e. $x \in \mathrm{sing}(\|V\|)\cap B_1$, we have

\[ \limsup_{r \to 0} \frac{\mathcal{H}^l_\infty(\text{sing}\|V\|\cap B_r(x))}{\omega_n r^l}\gt{}0. \]

This is equivalent to $\mathcal{H}^l(A)\gt{}0 \equiv \mathcal{H}^l_\infty(A)\gt{}0$ for any $A \subset \mathbb{R}^n$.

We choose $r_i \to 0$ such that

\[ \lim_{i \to \infty} \frac{\mathcal{H}^l_\infty(\text{sing}\|V\|\cap B_{r_i}(x))}{\omega_n r_i^l}\gt{}0. \]

By taking a subsequence, we can assume $(\eta_{x,r_i})_{\#}V$ converges to $\boldsymbol{C} \in \mathrm{VarTan}(V,x)$.

If $\mathcal{H}^l_\infty(\mathrm{sing}\,\|\boldsymbol{C}\|)=0$, then for any $\varepsilon\gt{}0$, we can find a covering of $\mathrm{sing}\,\|\boldsymbol{C}\|$ by balls $\{ B_{s_j}(y_j) \}_{j=1}^\infty$ such that

\[ \sum_{j=1}^\infty s_j^l \lt{} \varepsilon. \]

Note that $\mathrm{sing}\,\|\boldsymbol{C}\|\cap B_1(0)$ is compact, we know $\mathrm{sing}\,\|(\eta_{x,r_i})_{\#}V\|\cap B_1(0)$ can also be covered by $\{ B_{s_j}(y_j) \}_{j=1}^\infty$ for $i$ large enough. Thus, we have

\[ \frac{\mathcal{H}^l_\infty(\mathrm{sing}\,\|V\|\cap B_{r_i}(x))}{r_i^l}\lt{}\varepsilon \]

for $i$ large enough, which is a contradiction. ◻

Now we apply the above proposition iteratively to obtain a sequence of varifolds $\{ V_k \}_{k=0}^{K}$ such that:

$V_0=V$.
$V_{k+1} \in \mathrm{VarTan}(V_k,x_k)$ for some $x_k \in \mathrm{sing}(\|V_k\|)\cap B_1 \backslash \mathcal{S}(V_k)$.
$\operatorname{dim}(\mathcal{S}(V_{k+1}))\gt{}\operatorname{dim}(\mathcal{S}(V_k))$ for each $0\leq k \leq K-1$.
$\mathcal{H}^l(\mathrm{sing}(\|V_k\|)\cap B_1)\gt{}0$ for each $0\leq k \leq K$.
$V_K=\boldsymbol{C}\times \mathbb{R}^{m}$ where $\boldsymbol{C}\backslash \{0\}$ is a smooth immersed cone for some $m\geq 0$.

In particular, the last two conditions imply $m=l$. Now, $\boldsymbol{C}$ becomes a stable cone with isolated singular point $0$ of dimension $n-l$. By the classification of the stable cones, we know $n-l\geq 7$, and hence $l\leq n-7$. This shows $\operatorname{dim}(\mathrm{sing}\|V\|\cap B_1)\leq n-7$.

In the case $n=7$, we have

\[ \mathcal{H}^\alpha(\mathrm{sing}\,\|V_K\|\cap B_1)=0 \]

for any $\alpha\gt{}0$ by the above result.

If $\mathrm{sing}(\|V\|\cap B_1)$ is not discrete, then we can find $x_j \in \mathrm{sing}(\|V\|\cap B_1)$ such that $x_j \to x_0 \in \mathrm{sing}(\|V\|\cap B_1)$. Now, up to a subsequence, we can assume $(\eta_{x_0,|x_j-x_0|})_{\#}V$ converges to $\boldsymbol{C} \in \mathrm{VarTan}(V,x_0)$ and we denote $\xi = \lim_{j \to \infty} \frac{x_j-x_0}{|x_j-x_0|} \neq 0$. So $\mathcal{S}(\boldsymbol{C})$ contains the line spanned by $\xi$. In particular, $\mathcal{H}^1(\mathrm{sing}(\|\boldsymbol{C}\|)\cap B_1)\gt{}0$ which is a contradiction. Hence, $\mathrm{sing}(\|V\|\cap B_1)$ is discrete. ◻

First and Second Variation Formulas

Mon, 29 Jun 2026 00:00:00 +0000

We consider a variation of the submanifold $\Sigma$ given by a family of immersions $\iota_t: \Sigma \to M$ with $\iota_0 = \iota$. The variation vector field is defined as

\[ V = \frac{\partial \iota_t}{\partial t}\bigg|_{t=0}. \]

The volume element of $\Sigma_t = \iota_t(\Sigma)$ is denoted by $d\Sigma_t$.

Suppose $U \subset \Sigma$ is an open subset. We consider the variation of the volume functional

\[ \mathcal{A}(t)=|\iota_t(U)| = \int_{U} d\Sigma_t. \]

Proposition 1.3.1

The first variation of the volume functional is given by

\[ \delta_V(U):=\mathcal{A}'(0) = \int_{ \Sigma}\mathrm{div}^\Sigma V \, d\Sigma. \]

The second variation of the volume functional is given by

\[ \begin{aligned} \delta^{2}_V(U):={}\mathcal{A}''(0) = \int_\Sigma \sum_{i=1}^n-\left< R(V,e_i)V,e_i \right> + \mathrm{div}^\Sigma \nabla_{V}V + (\mathrm{div}^\Sigma V)^{2} \\ + \sum_{i=1}^{n}|(\nabla_{e_i}V)^\bot|^{2} - \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right> \, d\Sigma. \end{aligned} \]

Note that we do not impose any conditions on $V$ and $\Sigma$ for the above formulas. In particular, $\Sigma$ does not need to be a critical point of the volume functional, and $V$ does not need to be a normal variation.

Suppose $\{e_i\}$ is an orthonormal basis of $T\Sigma$, and define $e_i(t)=\iota_{t*} e_i$. Then the area element can be expressed as

\[ d\Sigma_t = \sqrt{\det(\left< e_i(t),e_j(t) \right>)} \, d\Sigma, \]

where $d\Sigma$ is the area element of $\Sigma$. We need the following lemma:

Lemma 1.3.2

We have

\[ \mathrm{det}(I+tA) = 1 + t \cdot \mathrm{tr}(A) + \frac{t^{2}}{2}(\mathrm{tr}(A)^{2} - \mathrm{tr}(A^{2})) + O(t^{3}), \]

Proof. We recall the standard formula for the determinant of a matrix $M$:

\[ \mathrm{det}(M) = \exp(\mathrm{tr}(\log M)). \]

Applying this to $I+tA$, we have

\[ \begin{aligned} \mathrm{det}(I+tA) &= \exp(\mathrm{tr}(\log(I+tA))) \\ &= \exp\left(\mathrm{tr}\left(tA - \frac{t^{2}}{2}A^{2} + O(t^{3})\right)\right) \\ &= \exp\left(t \cdot \mathrm{tr}(A) - \frac{t^{2}}{2} \mathrm{tr}(A^{2}) + O(t^{3})\right) \\ &= 1 + t \mathrm{tr}(A) + \frac{t^{2}}{2}(\mathrm{tr}(A)^{2} - \mathrm{tr}(A^{2})) + O(t^{3}). \end{aligned} \]

◻

We thus obtain the first variation formula:

\[ \begin{aligned} &\frac{d}{dt}|_{t=0} |\Sigma_t\cap U| = \frac{d}{dt}|_{t=0} \int_{\Sigma} \sqrt{\det(\left< e_i,e_j \right>)} \, d\Sigma \\ ={}& \frac{1}{2} \int_{\Sigma} \mathrm{tr}(\left< e_i',e_j \right> + \left< e_i,e_j' \right> )_{ij} \, d\Sigma = n \int_{\Sigma} H \, d\Sigma\\ ={}& \int_{\Sigma} \left< \nabla_{e_i}V,e_i \right> d\Sigma=\int_{ \Sigma} \mathrm{div}^\Sigma V d\Sigma \end{aligned} \]

For the second derivative of the area element, set $M(t)=(\left< e_i(t),e_j(t) \right>)_{ij}$. Then

\[ \begin{aligned} &\frac{d^2}{dt^2}|_{t=0} \sqrt{\mathrm{det}(M(t))}=\frac{1}{2}\frac{(\mathrm{det}\,M(t))''}{\sqrt{\mathrm{det}\,M(t)}}-\frac{1}{4}\frac{(\mathrm{det}\,M(t))'^2}{(\mathrm{det}\,M(t))^{3/2}}\\ ={}&\frac{1}{2}(\mathrm{tr}\,M''+(\mathrm{tr}\,M')^{2}-\mathrm{tr}\,(M'^{2}))-\frac{1}{4}(\mathrm{tr}\,M')^{2}\\ ={}&\frac{1}{2}\mathrm{tr}\,M''+\frac{1}{4}(\mathrm{tr}\,M')^{2}-\frac{1}{2}\mathrm{tr}\,(M'^{2}). \end{aligned} \]

We compute each term:

\[ \begin{aligned} \frac{1}{2}\mathrm{tr}\,M''={}&\sum_{i=1}^n \left< \nabla_{V}\nabla_{V}e_i,e_i \right> + \sum_{i=1}^n \left< \nabla_{V}e_i,\nabla_{V}e_i \right>\\ ={}&\sum_{i=1}^n-\left< R(V,e_i)V,e_i \right> + \left< \nabla_{e_i}\nabla_{V}V,e_i \right> + \sum_{i=1}^{n}\left< \nabla_{e_i}V,\nabla_{e_i}V \right> \\ ={}&\sum_{i=1}^n-\left< R(V,e_i)V,e_i \right> + \mathrm{div}^\Sigma \nabla_{V}V + \sum_{i=1}^{n}|\nabla_{e_i} V|^{2},\\ \frac{1}{4}(\mathrm{tr}\,M')^{2}={}&\frac{1}{4}\left(\sum_{i=1}^n \left< \nabla_{V}e_i,e_i \right> + \left< e_i,\nabla_{V}e_i \right> \right)^{2}=\left(\sum_{i=1}^n \left< \nabla_{e_i}V,e_i \right> \right)^{2}=(\mathrm{div}^\Sigma V)^{2},\\ \frac{1}{2}\mathrm{tr}\,(M'^{2})={}&\frac{1}{2}\sum_{i,j=1}^n \left( \left< \nabla_{e_i}V, e_j\right> + \left< \nabla_{e_j}V, e_i \right> \right) ^{2} = \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> ^{2} + \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right>\\ ={}&\sum_{i}^n |(\nabla_{e_i}V)^\top|^{2} + \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right>. \end{aligned} \]

\[ \begin{aligned} \frac{d^2}{dt^2}|_{t=0} \sqrt{\mathrm{det}(M(t))} = \sum_{i=1}^n-\left< R(V,e_i)V,e_i \right> + \mathrm{div}^\Sigma \nabla_{V}V + (\mathrm{div}^\Sigma V)^{2}\\ + \sum_{i=1}^{n}|(\nabla_{e_i}V)^\bot|^{2} - \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right>. \end{aligned} \]

Then, we have the second variation formula:

\[ \begin{aligned} &\frac{d^2}{dt^2}|_{t=0} |\Sigma_t\cap U| = \int_\Sigma \frac{d^2}{dt^2}|_{t=0} \sqrt{\mathrm{det}(M(t))} \, d\Sigma \\ ={}&\int_\Sigma \sum_{i=1}^n-\left< R(V,e_i)V,e_i \right> + \mathrm{div}^\Sigma \nabla_{V}V + (\mathrm{div}^\Sigma V)^{2} \\ & + \sum_{i=1}^{n}|(\nabla_{e_i}V)^\bot|^{2} - \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right> \, d\Sigma. \end{aligned} \]

If $V$ has compact support in $U$, then by the divergence theorem

\[ \begin{aligned} &\int_\Sigma \mathrm{div}^\Sigma V \, d\Sigma = \int_{ \Sigma}\mathrm{div}^\Sigma (V^\bot+V^\top) d\Sigma\\ ={}&\int_{ \Sigma}\mathrm{div}^\Sigma V^\bot d\Sigma = \int_{ \Sigma} \left< \vec{H},V \right> d\Sigma. \end{aligned} \]

Definition 1.3.3

A submanifold $\Sigma$ is called minimal if it is a critical point of the volume functional, i.e., $\delta_V(U) = 0$ for all variations $V$ with compact support in $U$.

By the first variation formula, $\Sigma$ is minimal if and only if $\vec{H} = 0$.

Suppose $\Sigma$ is minimal, and the variation vector field $V$ is normal, i.e., $V = V^\perp$, with compact support in $U$. Then, we know $\mathrm{div}^\Sigma V = 0$, and $\int_{ \Sigma} \mathrm{div}^\Sigma \nabla_{V}V \, d\Sigma = 0$. So the second variation formula simplifies to

\[ \frac{d^{2}}{dt^{2}}\bigg|_{t=0} |\Sigma_t\cap U| = \int_\Sigma \sum_{i=1}^n-\left< R(V,e_i)V,e_i \right> + \sum_{i=1}^{n}|(\nabla_{e_i}V)^\bot|^{2} - \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right> \, d\Sigma. \]

Furthermore, if $\Sigma$ is a two-sided hypersurface, then we can write $V = f\nu$ for some smooth function $f$ with compact support in $U$. In this case, the second variation formula becomes

\[ \frac{d^{2}}{dt^{2}}\bigg|_{t=0} |\Sigma_t\cap U| = \int_\Sigma |\nabla f|^{2} - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} \, d\Sigma, \]

This formula is useful in the study of minimal hypersurfaces. For instance, it directly yields the second variation formula for minimal hypersurfaces with free boundary.

Proposition 1.3.4

Suppose $(\Sigma,\partial\Sigma)\hookrightarrow (M,\partial M)$ is a minimal hypersurface with free boundary, i.e., $\Sigma$ meets $\partial M$ orthogonally along $\partial \Sigma$. Then the second variation formula for normal variations $V = f\nu$ with compact support in $U$ is

\[ \frac{d^{2}}{dt^{2}}\bigg|_{t=0} |\Sigma_t\cap U| = \int_\Sigma |\nabla f|^{2} - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} \, d\Sigma - \int_{\partial \Sigma} A_{\partial M}(\nu,\nu)f^{2} \, d\sigma, \]

Proof. It remains only to handle the term $\int_{ \Sigma} \mathrm{div}^\Sigma \nabla_{V}V \, d\Sigma$ in the second variation formula. By the divergence theorem, we have

\[ \int_{ \Sigma} \mathrm{div}^\Sigma \nabla_{V}V \, d\Sigma = \int_{\partial \Sigma} \left< \nabla_{V}V, \eta \right> \, d\sigma=\int_{\partial \Sigma} A_{\partial M}(\nu,\nu)f^{2} \, d\sigma, \]

◻

Definition 1.3.5

A minimal hypersurface $\Sigma$ is called stable if the second variation of the volume functional is nonnegative for all variations with compact support, i.e., $\frac{d^{2}}{dt^{2}}\bigg|_{t=0} |\Sigma_t\cap U| \geq 0$ whenever $\mathrm{spt}\,V \subset U$.

If $\Sigma$ is a two-sided stable minimal hypersurface, this is equivalent to the following stability inequality for all smooth functions $f$ with compact support in $U$:

\[ \int_\Sigma |\nabla f|^{2} - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} \, d\Sigma \geq 0. \]

Corollary 1.3.6

The variation of mean curvature $H$ is given by

\[ \frac{d}{dt}H=-\Delta f - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f, \]

where $V=f\nu$ is a normal variation with compact support in $U$.

Proof. We consider only the normal variation $V = f\nu$. The first variation gives

\[ \int_{ \Sigma} H f d\Sigma. \]

Hence the second variation can be expressed as

\[ \int_{ \Sigma} H \frac{d}{dt}f + \frac{d}{dt}H f + (Hf)^{2} d\Sigma. \]

Comparing this expression with the second variation formula, we conclude that

\[ \begin{aligned} \int_{ \Sigma} \frac{d}{dt}H f={}&\int_{ \Sigma} -\sum_{i=1}^{n}\left< R(V,e_i)V,e_i \right> + \sum_{i=1}^{n}|(\nabla_{e_i}V)^\bot|^{2} - \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right>\\ ={}&\int_{ \Sigma} |\nabla f|^{2}- (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} d\Sigma\\ ={}&\int_{ \Sigma} -f\left( \Delta f + (|A|^{2} + \mathrm{Ric}(\nu,\nu))f \right) d\Sigma \end{aligned} \]

Since this holds for all $f$ with compact support in $U$, we have

\[ \frac{d}{dt}H =-\Delta f - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f. \]

◻

Define the Jacobi operator $L$ by

\[ L = -\Delta - (|A|^{2} + \mathrm{Ric}(\nu,\nu)). \]

Then the stability condition can be written as

\[ \int_\Sigma f L f \, d\Sigma \geq 0 \quad \text{for all } f \text{ with compact support in } U. \]

Definition 1.3.7

The Morse index of a minimal hypersurface $\Sigma$ is the dimension of the space of smooth functions $f$ with compact support in $U$ such that

\[ \int_\Sigma f L f \, d\Sigma \lt{} 0, \]

i.e., the maximum dimension of a subspace on which the quadratic form is negative definite.

Equivalently, the Morse index counts the number of negative eigenvalues (with multiplicity) of the Jacobi operator $L$ under appropriate boundary conditions. A minimal hypersurface is stable if and only if its Morse index is zero.

Theorem 1.3.8

Let $\Sigma^n$ be a complete minimal hypersurface in a Riemannian manifold $(M^{n+1}, g)$. If $\Sigma$ has finite Morse index, then $\Sigma$ is stable outside a compact set, i.e., there exists a compact set $K \subset \Sigma$ such that

\[ \int_\Sigma |\nabla f|^{2} - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} \, d\Sigma \geq 0 \]

for all $f \in C_c^\infty(\Sigma \setminus K)$.

Proof. Let $\text{ind}(\Sigma) = m \lt{} \infty$ be the Morse index of $\Sigma$. By definition, there exists a finite-dimensional subspace $V \subset C_c^\infty(\Sigma)$ of dimension $m$ such that the quadratic form

\[ Q(f) = \int_\Sigma |\nabla f|^{2} - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} \, d\Sigma \]

is negative definite on $V$, and $Q(f) \geq 0$ for all $f \perp V$ (in the $L^2$ inner product sense).

Since each function $f_i$ in a basis of $V$ has compact support, there exists a compact set $K \subset \Sigma$ such that $\text{spt}(f_i) \subset K$ for all $i = 1, \ldots, m$.

Now, for any $f \in C_c^\infty(\Sigma \setminus K)$, we have $\text{spt}(f) \cap K = \emptyset$. This means $f$ is orthogonal to every basis function $f_i$ of $V$ (since their supports are disjoint). Therefore, $f \perp V$, and by the definition of finite Morse index, we have

\[ Q(f) = \int_\Sigma |\nabla f|^{2} - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} \, d\Sigma \geq 0. \]

This proves that $\Sigma$ is stable outside the compact set $K$. ◻

Geroch’s Conjecture by Dimension Reduction

Mon, 29 Jun 2026 00:00:00 +0000

The following theorem is the form of the Geroch conjecture used in the positive mass theorem. The dimension restriction comes only from the regularity theory for area-minimizing hypersurfaces: an area-minimizing hypersurface in an $n$-manifold is smooth when $n\leq7$.

Theorem 6.3.1

Let $3\leq n\leq7$. Let $X^n$ be a closed oriented smooth manifold. If there is a continuous map

\[ F:X\to \mathbb{T}^n \]

of nonzero degree, then $X$ admits no metric of positive scalar curvature.

Proof. We argue by induction on $n$. The base $n=2$ is Gauss–Bonnet: if a closed oriented surface maps to $\mathbb{T}^2$ with nonzero degree, then it has genus at least one, so it cannot carry a metric with positive Gaussian curvature.

Assume now $3\leq n\leq7$, and suppose the theorem is known in dimension $n-1$. Let $g$ be a positive-scalar-curvature metric on $X$. Write

\[ \theta_1,\ldots,\theta_n\in H^1(\mathbb{T}^n;\mathbb{Z}) \]

for the standard degree-one cohomology classes, and set

\[ \omega_i:=F^*\theta_i\in H^1(X;\mathbb{Z}). \]

Since $\deg F\neq0$,

\[ \omega_1\smile\cdots\smile\omega_n\neq0 \qquad\text{in }H^n(X;\mathbb{Z}). \]

Let $\alpha=\omega_n$. Its Poincare dual is a nonzero integral homology class in $H_{n-1}(X;\mathbb{Z})$. Choose an area-minimizing integral current $\Sigma$ in this class. Since $n\leq7$, regularity theory implies that $\Sigma$ is a smooth embedded closed hypersurface, possibly with several components and integer multiplicities. It is oriented, two-sided, minimal, and stable.

By Proposition 6.2.1, every component of $\Sigma$ of dimension at least three admits a positive-scalar-curvature metric; in the surface case, Remark 6.2.2 says that every component is a two-sphere.

Now compute the cup product on $\Sigma$. Since $[\Sigma]$ is Poincare dual to $\omega_n$,

\[ \int_\Sigma \omega_1\smile\cdots\smile\omega_{n-1} = \int_X \omega_1\smile\cdots\smile\omega_n \neq0. \]

Therefore at least one connected component $\Sigma_0$ has

\[ \int_{\Sigma_0} \omega_1\smile\cdots\smile\omega_{n-1}\neq0. \]

Equivalently, the map

\[ F_0:=(F_1,\ldots,F_{n-1})|_{\Sigma_0}: \Sigma_0\to\mathbb{T}^{n-1} \]

has nonzero degree.

If $n-1=2$, this contradicts the fact that $\Sigma_0$ is a two-sphere. If $n-1\geq3$, then $\Sigma_0$ carries a positive-scalar-curvature metric by conformal descent, contradicting the induction hypothesis in dimension $n-1$. This proves the theorem. ◻

Corollary 6.3.2

For $3\leq n\leq7$ and for every closed oriented $n$-manifold $Y$, the connected sum

\[ \mathbb{T}^n\#Y \]

admits no metric of positive scalar curvature.

Proof. There is a degree-one map

\[ \mathbb{T}^n\#Y\to\mathbb{T}^n \]

obtained by collapsing $Y$ minus a ball to the connected-sum point and using the identity map on the torus side. Theorem 6.3.1 applies. ◻

Green-kernel proof of the stable Bernstein theorem in R^4

Mon, 29 Jun 2026 00:00:00 +0000

The Green-kernel proof of the stable Bernstein theorem in $\mathbb{R}^4$ is due to Cabré–Catino–Mari–Mastrolia–Roncoroni [CCM+26]. The key point is that this proof does not use $\mu$-bubbles. Instead, it combines:

the positive supersolution furnished by stability;
the minimal positive Green kernel $\mathscr G$ of $-\Delta^M$;
a sharp pointwise estimate for $|\nabla\mathscr G|$ obtained from Bochner–Kato and a simple convexity lemma;
a weighted Schoen–Simon–Yau inequality;
a logarithmic cut-off in the variable $\mathscr G$.

Theorem 5.3.1

Let $M^3\hookrightarrow\mathbb{R}^4$ be a connected, complete, two-sided, stable minimal immersion. Then $M$ is an affine hyperplane.

Reduction to bounded curvature. It is enough to prove the theorem under the auxiliary assumption

\[ |A|\in L^\infty(M). \]

Indeed, if the theorem were false, then $|A|$ is nonzero somewhere. If $|A|$ is globally bounded, we are already in the bounded-curvature case. If not, one performs the standard point-picking argument on intrinsic balls $B_j^M(p_0)$: choose $q_j$ and a scale $\lambda_j=|A(q_j)|\to\infty$ so that, after replacing the immersion by

\[ F_j(x)=\lambda_j\bigl(F(x)-F(q_j)\bigr), \]

the rescaled hypersurfaces have $|A_j|(q_j)=1$ and uniformly bounded curvature on larger and larger intrinsic balls around $q_j$. Stability and minimality are scale invariant, so a smooth local compactness theorem gives a complete, two-sided, stable minimal limit in $\mathbb{R}^4$ with bounded second fundamental form and $|A|(0)=1$. The bounded-curvature case below would force this limit to be flat, a contradiction.

Green kernel and stability supersolution. Let now $M^n\hookrightarrow\mathbb{R}^{n+1}$ be complete, two-sided, stable, and minimal, with $n\geq3$, and assume $|A|\in L^\infty(M)$. We keep $n$ arbitrary until the last step. Stability means

\begin{equation} \tag{5.3.1} \label{eq:green-stability} \int_M |A|^2\phi^2\,d\mu \leq \int_M |\nabla\phi|^2\,d\mu, \qquad \phi\in \operatorname{Lip}_c(M). \end{equation}

Equivalently, the operator $-\Delta-|A|^2$ is nonnegative. By the positive solution characterization of nonnegative Schrödinger operators, there exists a positive $C^2$ function $u$ such that

\begin{equation} \tag{5.3.2} \label{eq:green-u-supersolution} \Delta u+|A|^2u\leq0 \qquad\text{on }M. \end{equation}

We next explain why the Green kernel exists. The needed potential-theoretic input is the following.

Definition 5.3.2

Let $(N^m,g)$ be a connected, complete, noncompact Riemannian manifold. We say that $N$ is non-parabolic if the Laplacian $-\Delta^N$ admits a positive minimal Green kernel: for some, equivalently for every, point $o\in N$ there is a function $G(o,\cdot)\gt{}0$ on $N\setminus\{o\}$ such that

\[ \Delta^N G(o,\cdot)=-\delta_o, \]

in the distributional sense, and $G$ is minimal among all positive fundamental solutions. If no such positive Green kernel exists, $N$ is called parabolic.

Equivalently, $N$ is parabolic if every positive superharmonic function on $N$ is constant; equivalently, the Brownian motion on $N$ is recurrent. Another useful equivalent criterion is the capacity criterion below.

Definition 5.3.3

For a compact set $K\Subset N$, define its $2$-capacity by

\[ \operatorname{Cap}_2(K) := \inf\left\{ \int_N |\nabla\phi|^2\,d\mu: \phi\in C_c^\infty(N),\quad \phi\geq1 \text{ on a neighborhood of }K \right\}. \]

Then $N$ is non-parabolic if and only if $\operatorname{Cap}_2(K)\gt{}0$ for some nonempty compact set $K$; it is parabolic if and only if $\operatorname{Cap}_2(K)=0$ for every compact $K$.

Proposition 5.3.4

Let $(N^m,g)$ be complete and noncompact, with $m\gt{}2$. Suppose that there is a constant $S\gt{}0$ such that

\begin{equation} \tag{5.3.3} \label{eq:sobolev-nonparabolic} \left(\int_N |f|^{\frac{2m}{m-2}}\,d\mu\right)^{\frac{m-2}{m}} \leq S\int_N |\nabla f|^2\,d\mu \qquad \forall f\in C_c^\infty(N). \end{equation}

Then $N$ is non-parabolic.

Proof. Choose a compact set $K\Subset N$ with $|K|\gt{}0$. If $\phi\in C_c^\infty(N)$ and $\phi\geq1$ near $K$, then

\[ |K| \leq \int_K |\phi|^{\frac{2m}{m-2}}\,d\mu \leq \int_N |\phi|^{\frac{2m}{m-2}}\,d\mu. \]

Raising both sides to the power $(m-2)/m$ and using (5.3.3), we get

\[ |K|^{\frac{m-2}{m}} \leq \left(\int_N |\phi|^{\frac{2m}{m-2}}\,d\mu\right)^{\frac{m-2}{m}} \leq S\int_N |\nabla\phi|^2\,d\mu. \]

Taking the infimum over all admissible $\phi$ gives

\[ \operatorname{Cap}_2(K) \geq S^{-1}|K|^{\frac{m-2}{m}}\gt{}0. \]

By the capacity criterion, $N$ is non-parabolic. ◻

Proposition 5.3.5

Let $M^n\hookrightarrow\mathbb{R}^{n+1}$ be a complete minimal hypersurface with $n\gt{}2$. Then $M$ is non-parabolic.

Proof. We use the sharp Michael–Simon Sobolev inequality of Brendle [Bre21]; in the present minimal hypersurface case it gives the following $W^{1,2}$ Sobolev consequence:

\begin{equation} \tag{5.3.4} \label{eq:minimal-W12-sobolev} \left(\int_M |f|^{\frac{2n}{n-2}}\,d\mu\right)^{\frac{n-2}{n}} \leq C_n \int_M |\nabla f|^2\,d\mu . \end{equation}

Since $n\gt{}2$, Proposition 5.3.4 applies and $M$ is non-parabolic. ◻

For the theorem, Proposition 5.3.5 applies to our $M^n$, since $n\geq3$. Hence there exists a minimal positive Green kernel $\mathscr G$ of $-\Delta$ with pole $o\in M$:

\[ \Delta\mathscr G=-\delta_o,\qquad \mathscr G\gt{}0,\qquad \mathscr G(x)\to0\quad\text{as }x\to\infty. \]

For a.e. $s\gt{}0$, the level set

\[ \Sigma_s:=\{\mathscr G=s\}, \]

is smooth, and the Green flux identity gives

\begin{equation} \tag{5.3.5} \label{eq:green-flux} \int_{\Sigma_s}|\nabla\mathscr G|\,d\sigma=1. \end{equation}

Indeed, applying the divergence theorem to $\{\mathscr G\gt{}s\}\setminus B_\varepsilon(o)$ and letting $\varepsilon\to0$ gives exactly the mass of the pole.

Ricci lower bound from the Gauss equation. Let $\{e_i\}_{i=1}^n$ diagonalize $A$ at a point, with principal curvatures $\kappa_i$ and $\sum_i\kappa_i=0$. The traced Gauss equation gives

\[ \mathrm{Ric}(e_i,e_i) = \sum_{j\neq i}A_{ii}A_{jj} = -\kappa_i^2. \]

Thus, for every unit vector $v$,

\[ \mathrm{Ric}(v,v)=-|A(v,\cdot)|^2. \]

Since $A$ is trace-free,

\[ \kappa_i^2 = \left(\sum_{j\neq i}\kappa_j\right)^2 \leq (n-1)\sum_{j\neq i}\kappa_j^2 =(n-1)(|A|^2-\kappa_i^2), \]

and hence $\kappa_i^2\leq\frac{n-1}{n}|A|^2$. Therefore

\begin{equation} \tag{5.3.6} \label{eq:green-ricci-lower} \mathrm{Ric}\geq -\frac{n-1}{n}|A|^2g. \end{equation}

The key convexity lemma. We shall use the following elementary computation several times.

Lemma 5.3.6

Let $w,v\gt{}0$ be $C^2$ functions on a domain $\Omega\subset M$ satisfying

\[ \Delta w+Ww\geq0,\qquad \Delta v+Vv\leq0. \]

For $\delta\geq0$, set

\[ \xi_\delta=w^{1+\delta}v^{-\delta}. \]

Then

\begin{equation} \tag{5.3.7} \label{eq:green-magic} \Delta\xi_\delta \geq \bigl(\delta V-(1+\delta)W\bigr)\xi_\delta +\delta(1+\delta) \left|\nabla\log\frac{w}{v}\right|^2\xi_\delta . \end{equation}

Proof. Since $\xi_\delta=e^{\log\xi_\delta}$,

\[ \Delta\xi_\delta = \xi_\delta\left(\Delta\log\xi_\delta+ |\nabla\log\xi_\delta|^2\right). \]

Now

\[ \log\xi_\delta=(1+\delta)\log w-\delta\log v. \]

Using

\[ \Delta\log w=\frac{\Delta w}{w}-|\nabla\log w|^2, \qquad \Delta\log v=\frac{\Delta v}{v}-|\nabla\log v|^2, \]

we get

\[ \begin{aligned} \Delta\log\xi_\delta &=(1+\delta)\Delta\log w-\delta\Delta\log v \\ &\geq \delta V-(1+\delta)W -(1+\delta)|\nabla\log w|^2 +\delta|\nabla\log v|^2 . \end{aligned} \]

Also,

\[ \begin{aligned} |\nabla\log\xi_\delta|^2 &=(1+\delta)^2|\nabla\log w|^2 +\delta^2|\nabla\log v|^2 \\ &\quad -2\delta(1+\delta) \left\langle\nabla\log w,\nabla\log v\right\rangle . \end{aligned} \]

Adding the last two displays gives

\[ \Delta\log\xi_\delta+|\nabla\log\xi_\delta|^2 \geq \delta V-(1+\delta)W +\delta(1+\delta) \left|\nabla\log w-\nabla\log v\right|^2, \]

which is (5.3.7). ◻

Pointwise gradient estimate for the Green kernel. We first record the local differential inequality for the Green kernel.

Proposition 5.3.7

Let

\[ q=|\nabla\mathscr G|, \qquad \alpha=\frac{n-2}{n-1}, \qquad w=q^\alpha . \]

On the open set $\{q\gt{}0\}\setminus\{o\}$, the function $w$ satisfies

\begin{equation} \tag{5.3.8} \label{eq:green-gradient-subsolution} \Delta w+\frac{n-2}{n}|A|^2w\geq0. \end{equation}

Proof. Recall first the standard Bochner formula: for every smooth function $f$ on a Riemannian manifold,

\[ \frac12\Delta |\nabla f|^2 = |\nabla^2 f|^2 +\left\langle\nabla f,\nabla\Delta f\right\rangle +\mathrm{Ric}(\nabla f,\nabla f). \]

We apply this to $f=\mathscr G$ on $M\setminus\{o\}$. Since $\Delta\mathscr G=0$ away from the pole, the middle term vanishes. Hence

\[ \frac12\Delta q^2 = |\nabla^2\mathscr G|^2+\mathrm{Ric}(\nabla\mathscr G,\nabla\mathscr G). \]

The refined Kato inequality for harmonic functions gives

\[ |\nabla^2\mathscr G|^2\geq \frac{n}{n-1}|\nabla q|^2. \]

Together with the Ricci lower bound (5.3.6), this gives

\[ \frac12\Delta q^2 \geq \frac{n}{n-1}|\nabla q|^2 -\frac{n-1}{n}|A|^2q^2. \]

Since

\[ \frac12\Delta q^2=q\Delta q+|\nabla q|^2, \]

we subtract $|\nabla q|^2$ and obtain

\begin{equation} \tag{5.3.9} \label{eq:green-q-ineq} q\Delta q \geq \frac{1}{n-1}|\nabla q|^2 -\frac{n-1}{n}|A|^2q^2. \end{equation}

Now compute the Laplacian of $w=q^\alpha$:

\[ \begin{aligned} \Delta w &=\alpha q^{\alpha-1}\Delta q +\alpha(\alpha-1)q^{\alpha-2}|\nabla q|^2 \\ &\geq \alpha q^{\alpha-2} \left[ \frac{1}{n-1}|\nabla q|^2 -\frac{n-1}{n}|A|^2q^2 \right] +\alpha(\alpha-1)q^{\alpha-2}|\nabla q|^2, \end{aligned} \]

where we used (5.3.9) in the second line. The coefficient of $|\nabla q|^2$ is

\[ \alpha\left(\frac1{n-1}+\alpha-1\right) = \alpha\left(\frac1{n-1}+\frac{n-2}{n-1}-1\right)=0. \]

Hence

\[ \Delta w \geq -\alpha\frac{n-1}{n}|A|^2q^\alpha = -\frac{n-2}{n}|A|^2w, \]

which is exactly (5.3.8). ◻

Apply Lemma 5.3.6 to

\[ W=\frac{n-2}{n}|A|^2,\qquad V=|A|^2,\qquad v=u, \qquad \delta\geq0. \]

For

\[ \xi_\delta=w^{1+\delta}u^{-\delta}, \]

the coefficient of the zeroth-order $|A|^2\xi_\delta$ term in Lemma 5.3.6 is

\[ \delta V-(1+\delta)W = \left[ \delta-(1+\delta)\frac{n-2}{n} \right]|A|^2 = \frac{2\delta-(n-2)}{n}|A|^2 . \]

Thus this coefficient is nonnegative when $\delta\geq\frac{n-2}{2}$, and it vanishes exactly at the critical choice

\[ \delta=\frac{n-2}{2}. \]

With this choice, $1+\delta=\frac n2$, and

\[ \xi:=w^{n/2}u^{-(n-2)/2} = |\nabla\mathscr G|^{\frac{n(n-2)}{2(n-1)}}u^{-\frac{n-2}{2}} \]

is subharmonic on $\{q\gt{}0\}\setminus U$, where

\[ U:=\{\mathscr G\gt{}1\}. \]

Because $|A|$ is bounded, (5.3.6) gives a lower Ricci bound, and the Cheng–Yau gradient estimate [CY75] applied outside $U$ gives

\[ |\nabla\mathscr G|\leq C\mathscr G. \]

We package the remaining two comparison-principle arguments into one lemma. First $\mathscr G$ is compared with the stability supersolution $u$; this auxiliary estimate is used only to remove the negative power of $u$ and prove $\xi\to0$ at infinity. Then $\xi$ is compared directly with the harmonic barrier $\mathscr G$.

Lemma 5.3.8

There is a constant $C_0\gt{}0$ such that

\[ \xi\leq C_0\mathscr G \qquad\text{on }M\setminus U. \]

Proof. We first prove the auxiliary comparison

\[ \mathscr G\leq Cu \qquad\text{on }M\setminus U. \]

Since $\mathscr G\to0$ at infinity, the set $\bar U=\{\mathscr G\geq1\}$ is compact. Choose $C$ so large that

\[ \mathscr G\leq Cu \qquad\text{on }\partial U. \]

This is possible because $\mathscr G=1$ on $\partial U$ and $u\gt{}0$ there. Let $\Omega_R$ be a smooth exhaustion of $M$ with $\bar U\Subset\Omega_R$, and let $\mathscr G_R$ be the Dirichlet Green function on $\Omega_R$ with pole $o$. On the annular region $\Omega_R\setminus \bar U$, the function $\mathscr G_R$ is harmonic and vanishes on $\partial\Omega_R$. Also,

\[ \Delta(Cu)=C\Delta u\leq -C|A|^2u\leq0, \]

so $Cu$ is superharmonic. On the boundary of $\Omega_R\setminus \bar U$ we have

\[ \mathscr G_R\leq \mathscr G=1\leq Cu \quad\text{on }\partial U, \qquad \mathscr G_R=0\leq Cu \quad\text{on }\partial\Omega_R. \]

The comparison principle on the bounded annulus therefore gives

\[ \mathscr G_R\leq Cu \qquad\text{on }\Omega_R\setminus U. \]

Letting $R\to\infty$ and using $\mathscr G_R\uparrow\mathscr G$, we obtain

\[ \mathscr G\leq Cu \qquad\text{on }M\setminus U. \]

This is where the first comparison is used. Combining it with $|\nabla\mathscr G|\leq C\mathscr G$, we get

\[ \xi \leq C\mathscr G^{\frac{n(n-2)}{2(n-1)}}u^{-\frac{n-2}{2}} \leq C\mathscr G^{\frac{n-2}{2(n-1)}}\to0 \qquad\text{as }x\to\infty. \]

Now we perform the second comparison, this time between the subharmonic quantity $\xi$ and a multiple of the harmonic Green kernel $\mathscr G$. Since $\bar U$ is compact and $\xi$ is continuous up to $\partial U$, while $\mathscr G=1$ on $\partial U$, we can choose $C_0$ so large that

\[ \xi\leq C_0\mathscr G \qquad\text{on }\partial U. \]

Set

\[ h:=\xi-C_0\mathscr G . \]

On $M\setminus U$, the pole $o$ is not present, so $\mathscr G$ is harmonic. Also $\xi$ is subharmonic there, in the weak sense obtained from the preceding pointwise computation by the standard regularization $q_\tau=(q^2+\tau)^{1/2}$ and then letting $\tau\downarrow0$. Hence

\[ \Delta h=\Delta\xi-C_0\Delta\mathscr G\geq0 \qquad\text{on }M\setminus U. \]

Thus $h$ is subharmonic. Moreover,

\[ h\leq0\qquad\text{on }\partial U. \]

We now explain how the boundary condition at infinity is used. Since $\xi(x)\to0$ and $\mathscr G(x)\to0$ as $x\to\infty$, for every $\varepsilon\gt{}0$ there is a compact set $K_\varepsilon\supset \bar U$ such that

\[ h(x)=\xi(x)-C_0\mathscr G(x)\leq \xi(x)\leq\varepsilon \qquad\text{on }M\setminus K_\varepsilon . \]

Choose a smooth bounded domain $\Omega_\varepsilon$ with $K_\varepsilon\Subset\Omega_\varepsilon$. On the bounded annular domain $\Omega_\varepsilon\setminus\bar U$, the maximum principle gives

\[ h\leq\max\{0,\varepsilon\}=\varepsilon. \]

Indeed, the inner boundary gives $h\leq0$, and the outer boundary lies in $M\setminus K_\varepsilon$, where $h\leq\varepsilon$. Letting $\varepsilon\downarrow0$, we obtain $h\leq0$ on $M\setminus U$, namely $\xi\leq C_0\mathscr G$. ◻

Taking the power $2/n$ yields the Green-kernel gradient estimate

\begin{equation} \tag{5.3.10} \label{eq:green-gradient-estimate} |\nabla\mathscr G|^{\frac{n-2}{n-1}} \leq C u^{\frac{n-2}{n}}\mathscr G^{\frac2n} \qquad\text{on }M\setminus U. \end{equation}

Weighted Schoen–Simon–Yau inequality. Set

\[ \beta=\frac{n-2}{n}. \]

Simons’ identity and the refined Kato inequality for $A$ give, on $\{|A|\gt{}0\}$,

\begin{equation} \tag{5.3.11} \label{eq:green-simons-kato} |A|\Delta |A| \geq \frac2n|\nabla|A||^2-|A|^4. \end{equation}

Let $\widetilde w=|A|^\beta$. Then

\[ \begin{aligned} \Delta\widetilde w &=\beta |A|^{\beta-1}\Delta|A| +\beta(\beta-1)|A|^{\beta-2}|\nabla|A||^2 \\ &\geq \beta |A|^{\beta-2} \left[ \frac2n|\nabla|A||^2-|A|^4 \right] +\beta(\beta-1)|A|^{\beta-2}|\nabla|A||^2 . \end{aligned} \]

Again the gradient coefficient vanishes because

\[ \frac2n+\beta-1 = \frac2n+\frac{n-2}{n}-1=0. \]

Hence

\begin{equation} \tag{5.3.12} \label{eq:green-A-subsolution} \Delta\widetilde w+\frac{n-2}{n}|A|^2\widetilde w\geq0. \end{equation}

For $0\lt{}t\leq1$, (5.3.2) gives

\[ \Delta u^t = t u^{t-1}\Delta u+t(t-1)u^{t-2}|\nabla u|^2 \leq -t|A|^2u^t. \]

Apply Lemma 5.3.6 to

\[ w=\widetilde w,\qquad v=u^t,\qquad W=\frac{n-2}{n}|A|^2,\qquad V=t|A|^2. \]

For $\delta\geq0$, define

\[ z=|A|^{\beta(1+\delta)}u^{-t\delta}. \]

Then

\[ \Delta z \geq \left(t\delta-(1+\delta)\frac{n-2}{n}\right)|A|^2z. \]

Adding $|A|^2z$ to both sides,

\begin{equation} \tag{5.3.13} \label{eq:green-z-subsolution} \Delta z+|A|^2z \geq \gamma |A|^2z, \qquad \gamma:= \frac2n+\left(t-\frac{n-2}{n}\right)\delta . \end{equation}

Assume $\gamma\gt{}0$. Using (5.3.1) with test function $z\psi$ and integrating by parts gives

\[ \begin{aligned} 0 &\leq \int_M |\nabla(z\psi)|^2-|A|^2z^2\psi^2 \\ &= \int_M z^2|\nabla\psi|^2 -\int_M z\psi^2(\Delta z+|A|^2z). \end{aligned} \]

Together with (5.3.13),

\[ \gamma\int_M |A|^2z^2\psi^2 \leq \int_M z^2|\nabla\psi|^2. \]

Put

\[ m:=1+\beta(1+\delta). \]

Taking $\psi=\varphi^m$ and using Young’s inequality,

\begin{equation} \tag{5.3.14} \label{eq:green-weighted-ssy} \int_M |A|^{2m}u^{-2t\delta}\varphi^{2m} \leq C\int_M u^{-2t\delta}|\nabla\varphi|^{2m}, \qquad \varphi\in\operatorname{Lip}_c(M). \end{equation}

Indeed, the right-hand side before Young is

\[ C\int_M |A|^{2(m-1)}u^{-2t\delta}\varphi^{2m-2}|\nabla\varphi|^2, \]

and this is bounded by

\[ \frac12\int_M |A|^{2m}u^{-2t\delta}\varphi^{2m} + C\int_M u^{-2t\delta}|\nabla\varphi|^{2m}. \]

The parameter choice and the logarithmic cut-off. We now choose $\varphi=\eta(\mathscr G)$. By the coarea formula,

\begin{equation} \tag{5.3.15} \label{eq:green-coarea} \begin{aligned} \int_M u^{-2t\delta}|\nabla\varphi|^{2m} &= \int_M u^{-2t\delta}|\eta'(\mathscr G)|^{2m} |\nabla\mathscr G|^{2m} \\ &= \int_0^\infty |\eta'(s)|^{2m} \left[ \int_{\Sigma_s} u^{-2t\delta}|\nabla\mathscr G|^{2m-1}\,d\sigma \right]ds . \end{aligned} \end{equation}

The estimate (5.3.10) gives

\[ u^{-t\delta}|\nabla\mathscr G|^{\beta(1+\delta)} \leq C u^{-t\delta+ \frac{(n-1)(n-2)}{n^2}(1+\delta)} \mathscr G^{\frac{2(n-1)}{n^2}(1+\delta)}. \]

We therefore impose

\begin{equation} \tag{5.3.16} \label{eq:green-choice-1} t\delta= \frac{(n-1)(n-2)}{n^2}(1+\delta). \end{equation}

Then

\[ u^{-2t\delta}|\nabla\mathscr G|^{2m-1} = |\nabla\mathscr G| \left( u^{-t\delta}|\nabla\mathscr G|^{\beta(1+\delta)} \right)^2 \leq C|\nabla\mathscr G| \mathscr G^{\frac{4(n-1)}{n^2}(1+\delta)}. \]

On $\Sigma_s$ this becomes

\[ u^{-2t\delta}|\nabla\mathscr G|^{2m-1} \leq C|\nabla\mathscr G|s^{\frac{4(n-1)}{n^2}(1+\delta)}. \]

Using the flux identity (5.3.5), (5.3.15) becomes

\[ \int_M u^{-2t\delta}|\nabla\eta(\mathscr G)|^{2m} \leq C\int_0^\infty |\eta'(s)|^{2m} s^{\frac{4(n-1)}{n^2}(1+\delta)}\,ds. \]

For a logarithmic cut-off to close, we need

\begin{equation} \tag{5.3.17} \label{eq:green-choice-2} \frac{4(n-1)}{n^2}(1+\delta)\geq 2m-1 = 1+\frac{2(n-2)}{n}(1+\delta). \end{equation}

This inequality has a nonnegative solution $\delta$ only for $n=3$. In that case,

\[ \beta=\frac13,\qquad \delta=\frac72,\qquad t=\frac27, \qquad m=\frac52, \qquad \gamma=\frac12. \]

Then (5.3.14) reads

\begin{equation} \tag{5.3.18} \label{eq:green-final-ssy-n3} \int_M |A|^5u^{-2}\varphi^5 \leq C\int_M u^{-2}|\nabla\varphi|^5. \end{equation}

Also, the Green gradient estimate becomes

\[ |\nabla\mathscr G|^{1/2} \leq Cu^{1/3}\mathscr G^{2/3}. \]

Therefore

\[ u^{-1}|\nabla\mathscr G|^{3/2} \leq C\mathscr G^2, \]

and, on $\Sigma_s$,

\[ u^{-2}|\nabla\mathscr G|^4 = |\nabla\mathscr G| \left(u^{-1}|\nabla\mathscr G|^{3/2}\right)^2 \leq C|\nabla\mathscr G|s^4. \]

Hence

\[ \int_M u^{-2}|\nabla\eta(\mathscr G)|^5 \leq C\int_0^\infty |\eta'(s)|^5s^4\,ds. \]

For $R\gt{}1$, choose the logarithmic cut-off

\[ \eta_R(s)= \begin{cases} 0,&0\leq s\leq R^{-2},\\[2mm] 2+\dfrac{\log s}{\log R},&R^{-2}\lt{}s\lt{}R^{-1},\\[3mm] 1,&s\geq R^{-1}. \end{cases} \]

Then $|\eta_R'(s)|=(s\log R)^{-1}$ on $(R^{-2},R^{-1})$ and $0$ elsewhere. Putting $\varphi=\eta_R(\mathscr G)$ in (5.3.18),

\[ \begin{aligned} \int_{\{\mathscr G\geq R^{-1}\}} |A|^5u^{-2} &\leq C\int_M u^{-2}|\nabla\eta_R(\mathscr G)|^5 \\ &\leq C\int_{R^{-2}}^{R^{-1}} \frac{s^4}{s^5(\log R)^5}\,ds \\ &= \frac{C}{(\log R)^5}\int_{R^{-2}}^{R^{-1}}\frac{ds}{s} = \frac{C}{(\log R)^4}. \end{aligned} \]

Letting $R\to\infty$, the sets $\{\mathscr G\geq R^{-1}\}$ exhaust $M$ up to the end, and the right-hand side tends to zero. Hence

\[ |A|^5u^{-2}\equiv0. \]

Since $u\gt{}0$, we conclude $A\equiv0$. Thus the immersion is totally geodesic, and the connected complete image is an affine hyperplane in $\mathbb{R}^4$. This proves Theorem 5.3.1.

Regularity in the Immersed Setting

Mon, 29 Jun 2026 00:00:00 +0000

One of the key ingredients in the proof of Theorem 4.1.8 is the following $\varepsilon$-regularity theorem.

Theorem 4.3.1

Let $n\geq 3$. Suppose $M^n$ is a two-sided stable minimal hypersurface immersed in $B^{n+1}_4(0)$ and the singular set of $M$ satisfies $\bar{n}:=\mathrm{dim}(\mathrm{sing}M)\lt{} n-2-\frac{2(n-2)}{n}$. Additionally, assume $\mathcal{H}^n(M\cap B^{n+1}_4(0))\leq \Lambda$ for some $\Lambda \in (0,+\infty)$. Then, for any $\alpha \in (\frac{n-2}{n},\min \left\{ \frac{n-\bar{n}-2}{2},1 \right\})$, there exists $\varepsilon=\varepsilon(n,\bar{n},\alpha,\Lambda) \in (0,1)$ such that if

\[ \int_{ B^{n+1}_2(0)\cap M} |A|^{2\alpha}\leq \varepsilon, \]

then

\[ \sup_{B^{n+1}_{\frac{1}{2}}(0)\cap M}|A|^{2\alpha}\leq C \int_{ B^{n+1}_2(0)\cap M} |A|^{2\alpha} \]

for some constant $C=C(n,\bar{n},\Lambda,\alpha)$.

The above result relies on the following weak (intrinsic) Caccioppoli inequality.

Lemma 4.3.2

For any $\alpha \in (\frac{n-2}{n},\min \left\{ \frac{n-\bar{n}-2}{2} ,1\right\})$, and any locally Lipschitz function $\phi$ supported in $B^{n+1}_3(0)$, we have

\begin{equation} \tag{4.3.1} \begin{aligned} \int_{M\cap \{u\gt{}k\}} \left( 1-\frac{k}{u} \right){}&|\nabla u|^2\phi^2\leq C\int_{M\cap \{u\gt{}k\}} (u-k)^2|\nabla \phi|^2\nonumber\\+{}&Ck^2\int_{ M\cap \{u\gt{}k\}} \left( (u -k)^{\frac{2}{\alpha}}+k^{\frac{2}{\alpha}} \right) \phi^2, \label{eq:lemWeakCacci} \end{aligned} \end{equation}

where the constant $C=C(n,\bar{n}, \Lambda,\alpha)$. Here $u=|A|^\alpha$.

Proof. We first show that (4.3.1) holds for bounded locally Lipschitz $\phi$ with compact support in $B^{n+1}_4(0)$, vanishing in a neighborhood of $\mathrm{sing}M \cap B^{n+1}_4(0)$, and for any $\alpha \in (\frac{n-2}{n},1)$.

For such $\phi$ and $\alpha$, choose $\varphi = (|A|^\alpha-k)^+ \phi$ for $k\geq 0$. One checks that $((|A|^\alpha-k)^+)^2 \in C^1(M)\cap W^{2,\infty}_{\mathrm{loc}}(M)$. Hence we can insert $\varphi$ into the stability inequality.

We observe that

\[ \begin{aligned} \frac{1}{2}\Delta(|A|^\alpha-k)^2={}&\alpha\left( 1-\frac{k}{|A|^{\alpha}} \right)|A|^{2\alpha-2}|A|\Delta|A|\\ +&\alpha \left( \left( 1-\frac{k}{|A|^\alpha} \right)(\alpha-1)+\alpha \right)|A|^{2\alpha-2}|\nabla|A||^2. %+\alpha[(2\alpha-1)|A|^\alpha-k(\alpha-1)]|A|^{\alpha-2}|\nabla|A||^2 \end{aligned} \]

Using Simons’ inequality

\[ |A|\Delta |A|\geq\frac{2}{n}|\nabla|A||^2-|A|^4, \]

we obtain

\[ \begin{aligned} \int_{M} |\nabla \varphi|^2 ={}& \int_{M_{\gt{}k}} \alpha^2|A|^{2\alpha-2}|\nabla|A||^2\phi^2 +(|A|^\alpha-k)^2|\nabla \phi|^2 +\frac{1}{2}\left< \nabla (|A|^\alpha-k)^2, \nabla \phi^2 \right> \\ ={}& \int_{M_{\gt{}k}} \alpha^2|A|^{2\alpha-2}|\nabla|A||^2\phi^2 +(|A|^\alpha-k)^2|\nabla \phi|^2 -\frac{1}{2}\phi^2\Delta(|A|^\alpha-k)^2\\ ={}& \int_{M_{\gt{}k}} (|A|^\alpha-k)^2|\nabla\phi|^2 -\int_{M_{\gt{}k}} \alpha \left( 1-\frac{k}{|A|^\alpha} \right)|A|^{2\alpha-2}|A|\Delta|A|\phi^2\\ &+\alpha(1-\alpha)\left( 1-\frac{k}{|A|^\alpha} \right)|A|^{2\alpha-2}|\nabla|A||^2\phi^2\\ \leq{}& \int_{M_{\gt{}k}} ((|A|^\alpha-k)^2)|\nabla\phi|^2\\ &-\int_{M_{\gt{}k}} \alpha\left( \frac{2}{n}+\alpha-1 \right) \left( 1-\frac{k}{|A|^\alpha} \right)|A|^{2\alpha-2}|\nabla|A||^2\phi^2 +\int_{M_{\gt{}k}} \alpha |A|^{2\alpha+2}\phi^2 \end{aligned} \]

where $M_{\gt{}k}$ denotes $M\cap \{|A|^\alpha\gt{}k\}$. On the other hand, by stability, we have

\[ \int_{M} |\nabla\varphi|^2\geq \int_{M} |A|^2\varphi^2=\int_{M_{\gt{}k}} |A|^{2}(|A|^{\alpha}-k)^2\phi^2. \]

Now, we write $\delta:=\alpha - \frac{n-2}{n}\gt{}0$. Then the stability inequality gives

\[ \begin{aligned} & \delta\int_{M_{\gt{}k}} \alpha \left( 1-\frac{k}{|A|^\alpha} \right)|A|^{2\alpha-2}|\nabla|A||^2\phi^2\\ \leq{}& \int_{M_{\gt{}k}} ((|A|^\alpha-k)^2)|\nabla \phi|^2 +\int_{M_{\gt{}k}} \alpha |A|^{2\alpha}|A|^2\phi^2\\ &-\int_{M_{\gt{}k}} |A|^{2}(|A|^{\alpha}-k)^2\phi^2. \end{aligned} \]

Now, let $u=|A|^\alpha$. Then,

\begin{equation} \tag{4.3.2} \frac{\delta}{\alpha} \int_{M_{\gt{}k}} \left( 1-\frac{k}{u} \right)|\nabla u|^2\phi^2\leq \int_{M_{\gt{}k}} (u-k)^2|\nabla \phi|^2+\int_{M_{\gt{}k}}u^{\frac{2}{\alpha}}\left( \alpha u^2-(u-k)^2 \right) \phi^2. \label{eq:pfWeakPoincareExtraTerm} \end{equation}

Now we estimate $u^{\frac{2}{\alpha}}\left( \alpha u^2-(u-k)^2 \right)$.

\begin{equation} \tag{4.3.3} \begin{aligned} \alpha u^2-(u -k)^2={} & -(1-\alpha) (u -k)^2+2\alpha k(u-k)+\alpha k^2 \leq \frac{\alpha^2k^2}{1-\alpha}+\alpha k^2=\frac{\alpha}{1-\alpha} k^{2}. \label{eq:pfYoung} \end{aligned} \end{equation}

where we used Young’s inequality. By the trivial inequality $(x+y)^a\leq 2^a(x^a+y^a)$ for $x,y\geq 0$, $a\gt{}0$, we have

\[ u^{\frac{2}{\alpha}}\left(\alpha u^2-(u-k)^2\right)\leq 2^{\frac{2}{\alpha}}\frac{\alpha}{1-\alpha} \left( (u -k)^{\frac{2}{\alpha}}+k^{\frac{2}{\alpha}} \right)k^2. \]

Substituting this into (4.3.2), we obtain (4.3.1).

To complete the proof, we show by approximation that (4.3.1) holds for any bounded locally Lipschitz $\phi$ supported in $B_3(0)$, assuming $\alpha \in (\frac{n-2}{n},\min \left\{ \frac{n-\bar{n}-2}{2},1 \right\})$. Note that $\phi$ may be non-zero on the singular set of $M$.

We first derive a preliminary estimate on $|A|$.

Lemma 4.3.3

If $\mathcal{H}^{n-2}(\mathrm{sing}(M)\cap B^{n+1}_4(0))=0$, then we have $|A| \in L^2(B^{n+1}_{\frac{7}{2}}(0)\cap M)$ and

\[ \int_{M\cap B^{n+1}_\rho(x) } |A|^2\leq C\rho^{n-2}, \]

for any $x \in B^{n+1}_{\frac{7}{2}}(0)$ and $\rho \in (0,\frac{1}{4})$, where $C=C(\Lambda)$.

Proof. For each $\varepsilon\gt{}0$, we choose balls $\left\{ B^{n+1}_{r_i}(x_i) \right\}_{i=1}^N$ such that $\mathrm{sing}(M)\cap B^{n+1}_4(0)\subset \bigcup_{i=1}^N B^{n+1}_{r_i}(x_i)$ and $\sum_{i=1}^N r_i^{n-2}\leq \varepsilon$. We choose $\zeta_i$ to be a non-negative $C^1$ function such that $\zeta_i$ is supported outside of $B^{n+1}_{r_i}(x_i)$, $\zeta_i=1$ outside of $B^{n+1}_{2r_{i}}(x_i)$, and $|\nabla \zeta_i|\leq \frac{2}{r_i}$. Then, we define $\zeta_\varepsilon=\min_{1\leq i\leq N}\zeta_i$. We insert $\zeta_\varepsilon \phi$ into the stability inequality where $\phi$ is a non-negative locally Lipschitz function with compact support in $B^{n+1}_4(0)$. Then,

\[ \int_{M\cap B^{n+1}_4(0)} |A|^2\phi^2\zeta_\varepsilon^2\leq 2\int_{M\cap B^{n+1}_4(0)} \left|\nabla \phi\right|^2\zeta_\varepsilon^2+2\int_{M\cap B^{n+1}_4(0)}|\nabla \zeta_\varepsilon|^2\phi^2 \]

by Cauchy-Schwarz inequality. Note that

\[ \begin{aligned} \int_{M\cap B^{n+1}_4(0)} |\nabla \zeta_\varepsilon|^2\phi^2\leq{} & \|\phi\|_{L^\infty(B^{n+1}_4(0))}^2\sum_{i=1}^{N}\int_{M\cap B^{n+1}_{2r_i}(x_i)} |\nabla \zeta_i|^2\\ \leq{}& C\|\phi\|_{L^\infty(B^{n+1}_4(0))}^2\sum_{i=1}^{N}r_i^{n-2}\leq C\|\phi\|_{L^\infty(B^{n+1}_4(0))}^2\varepsilon \end{aligned} \]

which converges to $0$ as $\varepsilon\to 0^+$. Then, we have

\[ \int_{M} |A|^2\phi^2\leq 2\int_{M}|\nabla \phi|^2. \]

In particular, it implies $|A| \in L^2(B^{n+1}_{\frac{7}{2}}(0))$ if we choose $\phi\equiv 1$ on $B^{n+1}_{\frac{7}{2}}(0)$. Now, we choose $\phi$ supported on $B^{n+1}_{2\rho}(x)$, and equal to $1$ on $B^{n+1}_\rho(x)$, with $|\nabla \phi|\leq \frac{2}{\rho}$. Together with the monotonicity formula, we have

\[ \int_{M\cap B^{n+1}_\rho(x)} |A|^2\leq 2\int_{M\cap B^{n+1}_{2\rho}(x)\backslash B^{n+1}_\rho(x)} \frac{1}{\rho^2}\leq C\rho^{n-2}, \]

for some $C=C(\Lambda)$. ◻

The remaining part is similar to the proof of the previous lemma. Based on the assumption of $\bar{n}$ and $\alpha$, we know $\mathcal{H}^{n-2-2\alpha}(\mathrm{sing}M)=0$. Therefore, for any $\varepsilon\gt{}0$, there exist $B^{n+1}_{r_1}(x_1),B^{n+1}_{r_2}(x_2),\cdots , B^{n+1}_{r_N}(x_N)$ with $x_i\in B^{n+1}_{\frac{7}{2}}(0)$ and $0\lt{}r_i\lt{}\frac{1}{4}$ for each $1\leq i\leq N$, such that

\begin{equation} \tag{4.3.4} \mathrm{sing}M\cap B^{n+1}_3(0)\subset \bigcup_{i=1}^N B^{n+1}_{r_i}(x_i),\quad \text{ and }\quad \sum_{i =1}^{N}r_i^{n-2-2\alpha}\leq \varepsilon. \label{eq:pfSingCover} \end{equation}

We choose $\zeta_i$ and $\zeta_\varepsilon$ as in the proof of Lemma 4.3.3. For any locally Lipschitz $\phi$ with compact support in $B^{n+1}_3(0)$, $\zeta_\varepsilon \phi$ vanishes near $\mathrm{sing}M$, allowing us to use (4.3.1) with $\zeta_\varepsilon \phi$ in place of $\phi$. Thus, we have

\begin{equation} \tag{4.3.5} \begin{aligned} {} & \int_{M_{\gt{}k}} \left( 1-\frac{k}{u} \right)|\nabla u|^2\zeta_\varepsilon^2\phi^2\nonumber \\ \leq {}& C\int_{M_{\gt{}k}} (u-k)^2\zeta_\varepsilon^2|\nabla \phi|^2+Ck^2\int_{M_{\gt{}k}} \left( (u -k)^{\frac{2}{\alpha}}+k^{\frac{2}{\alpha}} \right) \zeta_\varepsilon^2\phi^2 + C \int_{M_{\gt{}k}} (u-k)^2|\nabla\zeta_\varepsilon|^2\phi^2, \label{eq:pfWeakPoincareApprox} \end{aligned} \end{equation}

by the Cauchy-Schwarz inequality.

For the first two terms on the right-hand side of (4.3.5), since $|A|^{2\alpha}$ and $|A|^2$ are integrable in $B^{n+1}_3(0)\cap M$ by Lemma 4.3.3, we can let $\varepsilon\to 0^+$, leading to

\[ C \int_{M_{\gt{}k}} (u-k)^2|\nabla \phi|^2+Ck^2\int_{M_{\gt{}k}} \left( (u -k)^{\frac{2}{\alpha}}+k^{\frac{2}{\alpha}} \right) \phi^2. \]

Then, we need to show

\[ \lim_{\varepsilon\to 0^+} \int_{M_{\gt{}k}} (u-k)^2|\nabla\zeta_\varepsilon|^2\phi^2=0. \]

Applying Lemma 4.3.3, (4.3.4), and Hölder inequality, we obtain

\[ \begin{aligned} {} & \int_{M_{\gt{}k}} (u-k)^2|\nabla \zeta_\varepsilon|^2\phi^2\\ \leq {}& \sum_{i=1}^{N}\|\phi\|_{L^\infty(B^{n+1}_3(0))}^2\int_{M\cap B^{n+1}_{r_i}(x_i)} |A|^{2\alpha}|\nabla \zeta_i|^2\\ \leq{}& \|\phi\|_{L^\infty(B^{n+1}_3(0))}^2\sum_{i=1}^{N}\left( \int_{M\cap B^{n+1}_{r_i}(x_i)} |A|^2 \right)^{\alpha}\left( \int_{M\cap B^{n+1}_{r_i}(x_i)} |\nabla \zeta_i|^{\frac{2}{1-\alpha}} \right)^{1-\alpha}\\ \leq {}& C\|\phi\|_{L^\infty(B^{n+1}_3(0))}^2\sum_{i=1}^{N}r_i^{\alpha(n-2)}r_i^{\left(n-\frac{2}{1-\alpha}\right)(1-\alpha)}=C\|\phi\|_{L^\infty(B^{n+1}_3(0))}^2\sum_{i=1}^{N}r_i^{n-2-2\alpha}\\ \leq {}& C\|\phi\|_{L^\infty(B^{n+1}_3(0))}^2\varepsilon. \end{aligned} \]

Hence, letting $\varepsilon\to 0^+$, we conclude that (4.3.1) holds for any bounded locally Lipschitz $\phi$ supported in $B^{n+1}_3(0)$. ◻

The preceding proof also shows that $|\nabla u|^2$ is integrable in $B^{n+1}_3(0)\cap M$ and hence $u\in W^{1,2}(B^{n+1}_3(0)\cap M)$. We now prove Theorem 4.3.1.

Proof. Consider

\[ k_l=d\left( 1 - \frac{1}{2^{l-1}} \right),\quad \text{and}\quad R_l=\frac{1}{2}+\frac{1}{2^l}, \]

for $0\lt{}d\leq 1$. $k_l$ increases to $d$ and $R_l$ decreases to $1/2$ as $l\to \infty.$ For simplicity, we write $\Omega_l = M\cap \left\{ u\gt{}k_l \right\}\cap B^{n+1}_{R_l}(0)$.

Applying the previous lemma and noting that

\[ 1-\frac{k_l}{u}\geq 1-\frac{k_l}{k_{l+1}}\geq \frac{1}{2^l}, \]

for any $u\gt{}k_{l+1}$, we have

\[ \begin{aligned} \frac{1}{2^l}\int_{ M_{\gt{}k_{l+1}}} |\nabla u|^2 \phi^2\leq{}& C\left[ \int_{ M_{\gt{}k_{l}}} (u-k_{l})^2|\nabla \phi|^2\right.\\ &\left.+d^2\int_{ M_{\gt{}k_{l}}} (u -k_{l})^{\frac{2}{\alpha}}\phi^2+ d^{2+\frac{2}{\alpha}}\int_{ M_{\gt{}k_l}}\phi^2 \right] . \end{aligned} \]

Using

\[ |\nabla((u-k_{l+1})\phi)|^2\leq 2|\nabla u|^2\phi^2+2(u-k_{l+1})^2|\nabla \phi|^2, \]

we obtain

\[ \begin{aligned} {} & \int_{ M_{\gt{}k_{l+1}}} |\nabla ((u-k_{l+1})\phi)|^2\\ \leq {}& 2^l C\left[ \int_{ M_{\gt{}k_{l}}} (u-k_{l})^2|\nabla \phi|^2+d^2\int_{ M_{\gt{}k_{l}}} (u -k_{l})^{\frac{2}{\alpha}}\phi^2+ d^{2+\frac{2}{\alpha}}\int_{ M_{\gt{}k_l}}\phi^2 \right] . \end{aligned} \]

Now choose $\phi$ supported in $B^{n+1}_{R_l}(0)$, with $\phi=1$ on $B^{n+1}_{R_{l+1}}(0)$, $|\nabla \phi|\leq 2^{l+2}$, and $0\leq \phi \leq 1$. Together with the Michael–Simon inequality [MS73]

\[ \left( \int_{M} |\varphi|^{\frac{2n}{n-2}} \right)^{\frac{n-2}{n}}\leq C \int_{M} \left|\nabla \varphi\right|^2, \]

for a constant $C$ only depending on $n$. Then, we have

\begin{equation} \tag{4.3.6} \begin{aligned} {} & \left( \int_{ \Omega_{l+1}} (u-k_{l+1})^{\frac{2n}{n-2}} \right)^{\frac{n-2}{n}}\nonumber \\ \leq {}& C^l\left[ \int_{ \Omega_l} (u-k_{l})^2+d^2\int_{ \Omega_l} (u -k_{l})^{\frac{2}{\alpha}}+ d^{2+\frac{2}{\alpha}}\mathcal{L}^n(\Omega_l) \right]. \label{eq:pfIterEst} \end{aligned} \end{equation}

Using the fact that when $u\geq k_l$, we know $u-k_{l-1}\geq \frac{d}{2^{l-1}}$. Hence, for any $0\leq \beta\leq \frac{2n}{n-2}$,

\[ \begin{aligned} \int_{ \Omega_l} (u-k_l)^\beta\leq{} & \int_{ \Omega_l}(u-k_l)^\beta \left( \frac{2^{l-1}}{d} \right)^{\frac{2n}{n-2}-\beta}(u-k_{l-1})^{\frac{2n}{n-2}-\beta}\\ \leq{}& \frac{C^l}{d^{\frac{2n}{n-2}-\beta}}\int_{ \Omega_{l-1}} (u-k_{l-1})^{\frac{2n}{n-2}}, \end{aligned} \]

where constant $C=C(n)$. Note that since $\frac{2}{\alpha}\lt{} \frac{2n}{n-2}$, we can use the above inequality with $\beta=0,2$, and $\frac{2}{\alpha}$ in (4.3.6) to obtain

\begin{equation} \tag{4.3.7} S_{l+1}^{\frac{n-2}{n}}\leq C^l\left( \frac{1}{d^{\frac{2n}{n-2}-2}}+\frac{1}{d^{\frac{2n}{n-2}-\frac{2}{\alpha}-2}}+\frac{1}{d^{\frac{2n}{n-2}-2-\frac{2}{\alpha}}} \right)S_{l-1}, \label{eq:pfIterRaw} \end{equation}

where

\[ S_l:= \int_{\Omega_l} (u-k_{l})^{\frac{2n}{n-2}}. \]

Using $d\leq 1$, (4.3.7) implies

\begin{equation} \tag{4.3.8} \frac{S_{l+1}}{d^{\frac{2n}{n-2}}}\leq C^l \left( \frac{S_{l-1}}{d^{\frac{2n}{n-2}}} \right)^{\frac{n}{n-2}}, \label{eq:pfIter} \end{equation}

for some $C=C(n,\bar{n},\Lambda,\alpha)$. Iterating (4.3.8), we obtain

\[ \frac{S_{2l+1}}{d^{\frac{2n}{n-2}}}\leq C^{2+\frac{4(n-2)}{n}+\cdots + 2l (\frac{n-2}{n})^{l-1}}\left( C^2\frac{S_1}{d^{\frac{2n}{n-2}}} \right)^{\left( \frac{n}{n-2} \right)^l}\leq C^{\frac{n^2}{2}}\left( C^2\frac{S_1}{d^{\frac{2n}{n-2}}} \right)^{\left( \frac{n}{n-2} \right)^l}. \]

Hence, if we require

\[ S_1\leq (\varepsilon' d)^{\frac{2n}{n-2}}, \]

for some positive $\varepsilon'$ only depending on $n$, $\bar{n}$, $\delta$, $\Lambda$, and $\alpha$, then we have $\lim_{l\to \infty} S_{2l+1}=0$. This implies

\[ \left|A\right|^\alpha\leq d \text{ on }B^{n+1}_{\frac{1}{2}}(0). \]

Finally, we need to ensure $S_1\leq (\varepsilon' d)^{\frac{2n}{n-2}}$.

Using Lemma 4.3.2 with $k=0$ and a suitable test function, we obtain

\[ \int_{M\cap B^{n+1}_{\frac{3}{2}}(0)} |\nabla u|^2 \leq C\int_{M\cap B^{n+1}_2(0)}u^2 \]

for some $C=C(n,\bar{n},\Lambda,\alpha)$. Thus, by Michael–Simon’s inequality, we have

\begin{equation} \tag{4.3.9} S_1^{\frac{n-2}{n}}\leq C \int_{M\cap B^{n+1}_1(0)} \left|\nabla (u\varphi)\right|^2 \leq C \int_{M\cap B^{n+1}_2(0)} u^2=C\int_{M\cap B^{n+1}_2(0)} |A|^{2\alpha}. \label{eq:pfMSLastEst} \end{equation}

for $\varphi$ supported on $B^{n+1}_{\frac{3}{2}}(0)$, equal to $1$ on $B^{n+1}_1(0)$, and $|\nabla \varphi|\leq 4$, where $C=C(n,\bar{n},\Lambda,\alpha)$. Now choose $\varepsilon=\frac{(\varepsilon')^2}{C}$, where $C$ is the constant in (4.3.9), and set $d =\sqrt{\frac{1}{\varepsilon}\int_{M\cap B^{n+1}_2(0)} |A|^{2\alpha}} \in (0,1]$ by assumption. Consequently, $S_1^{\frac{n-2}{n}}\leq (\varepsilon' d)^2$ holds by (4.3.9). For such a choice of $d$, we know $\lim_{l\to \infty} S_l=0$, which implies

\[ \sup_{B^{n+1}_{\frac{1}{2}}(0)\cap M}|A|^{2\alpha}\leq d^2 =C \int_{M\cap B^{n+1}_2(0)\cap M} |A|^{2\alpha} \]

for some $C=C(n,\bar{n},\Lambda,\alpha)$. ◻

With this $\varepsilon$-regularity theorem, we can prove the following results.

Proposition 4.3.4

Let $n\geq 3$, and $\bar{n}\lt{}n-4+\frac{4}{n}$. Suppose $M_j$ is a sequence of immersed, two-sided, stable minimal hypersurfaces in $B^{n+1}_4(0)$ with $\mathrm{dim}(\mathrm{sing}(M_j)\cap B^{n+1}_4(0))\leq \bar{n}$, and that $M_j$ converges (as varifolds) to $q|P\cap B^{n+1}_4(0)|$ as $j\to \infty$, where $P$ is a hyperplane and $q$ is a positive integer. Then,

\[ \lim_{j\to \infty} \sup_{B^{n+1}_{\frac{1}{2}}(0)\cap M_j}|A_{M_j}|=0. \]

Moreover, $\mathrm{sing} M_j\cap B^{n+1}_{\frac{1}{4}}(0)=\emptyset$ and $M_j\cap B^{n+1}_{\frac{1}{4}}(0)$ has exactly $q$ connected components for $j$ large enough, and each component of $M_j\cap B^{n+1}_{\frac{1}{4}}(0)$ converges smoothly to $P$ in $B^{n+1}_{\frac{1}{4}}(0)$ as smooth immersions.

Proof. We suppose $P=\left\{ x_{n+1}=0 \right\}$.

We claim that

\[ \int_{ M_j\cap B_2(0)} |A_j|^{2\alpha} \to 0. \]

We also use the following theorem due to Schoen–Simon [SS81].

Theorem 4.3.5

We have the following inequality

\[ \int_{ } |A|^{2}\varphi^{2}\leq C \int_{ } |\nabla \varphi|^{2} (1-(\nu\cdot e_{n+1})^{2}) \]

By the monotonicity formula, we have

\[ \lim_{j\to \infty} \sup_{B^{n+1}_3(0)\cap M_j}|x_{n+1}|=0. \]

Otherwise, we can find a sequence $p_j \in M_j$ such that $|p_{j,n+1}|\geq \delta\gt{}0$ for some $0\lt{}\delta\lt{}1$. By the monotonicity formula, we have

\[ \mathcal{H}^n(M_j\cap B^{n+1}_{\frac{\delta}{2}}(p_j))\geq C \delta^n. \]

Then, we have

\[ q|P\cap B^{n+1}_4(0)|(B_{\delta}(p_0))\geq \limsup_{j \to \infty}V_j(B_{\frac{\delta}{2}}(p_j))\geq C \delta^n, \]

where $|p_{0,n+1}|\geq \delta$, a contradiction.

Now, we choose $\varphi^{2}x_{n+1}e_{n+1}$ as a test vector field in the first variation formula where $\varphi$ is a smooth function supported in $B^{n+1}_3(0)$. Then, we have

\[ \int_{ } \varphi^{2} |e_{n+1}^{\top}|^{2}=-2\int_{ } x_{n+1}\varphi \nabla \varphi \cdot e_{n+1}^{\top}\leq \frac{1}{2}\int_{ } \varphi^{2} |e_{n+1}^{\top}|^{2}+2\int_{ } |\nabla \varphi|^{2}|x_{n+1}|^{2}. \]

So we have

\[ \int_{ M_j} \varphi^{2}(1-(\nu\cdot e_{n+1})^{2})\leq 4\int_{ M_j} |\nabla \varphi|^{2}|x_{n+1}|^{2}, \]

which goes to zero as $j\to \infty$. Hence

\[ \int_{ M_j \cap B_2(0)}|A_j|^{2} \to 0. \]

Now, we use Hölder’s inequality to get

\[ \int_{ M_j \cap B_2(0)}|A_j|^{2\alpha}\leq \left( \int_{ M_j \cap B_2(0)}|A_j|^{2} \right)^{\alpha}\left( \mathcal{H}^n(M_j\cap B_2(0)) \right)^{1-\alpha}\to 0. \]

Now we apply the $\varepsilon$-regularity theorem (Theorem 4.3.1) to get

\begin{equation} \tag{4.3.10} \lim_{j\to \infty} \sup_{B^{n+1}_{\frac{1}{2}}(0)\cap M_j}|A_{M_j}|=0. \label{eq:pfLimitAconverges} \end{equation}

Next, let us denote $S_j=P(\mathrm{sing}(M_j))$ where $P$ is the projection to $\left\{ x_{n+1}=0 \right\}$. Then the projection $P$ gives a covering map from $M_j\cap (B_{\frac{1}{4}}^n(0)\setminus S_j) \times \mathbb{R}$ to $(B_{\frac{1}{4}}^n(0)\backslash S_j) \times \left\{ 0 \right\}$, and the covering degree is $q$ by (4.3.10) for $j$ large enough. Since $B_{\frac{1}{4}}^n(0)\backslash S_j$ is simply connected (because $\mathrm{dim}(S_j)\leq \bar{n}$), we know that $M_j\cap (B_{\frac{1}{4}}^n(0)\setminus S_j) \times \mathbb{R}$ has exactly $q$ connected components, and each component can be written as a graph of a smooth function over $B_{\frac{1}{4}}^n(0)\backslash S_j$. By the removable singularity theorem (cf. [DGS65, Sim77]), we know that such a function can be extended to a smooth function on $B_{\frac{1}{4}}^n$ which solves the minimal surface equation. Hence, $\mathrm{sing}M_j \cap B_{\frac{1}{4}}^n(0)\times \mathbb{R}=\emptyset$ and it can be decomposed into $q$ connected components, each of which converges smoothly to $B_{\frac{1}{4}}^n(0)\times \left\{ 0 \right\}$ as smooth immersions by standard PDE theory. ◻

We consider a flat cone $\boldsymbol{C}$ in $\mathbb{R}^{n+1}$ defined as a union of hyperplanes and half-hyperplanes. Explicitly, we write

\[ \boldsymbol{C}:=\sum_{i =1}^{N_1}p_i|P_i|+\sum_{i=1}^{N_2}q_i|H_i|, \]

for $\{ p_i \}_{i=1}^{N_1}$, $\{ q_i \}_{i=1}^{N_2}\subset \mathbb{N}$. Here $\{ P_i \}_{i=1}^{N_1}$ are distinct hyperplanes and $\{ H_i \}_{i=1}^{N_2}$ are distinct half-hyperplanes such that $0 \in P_i$ for each $1\leq i\leq N_1$, $0 \in \bar{H}_i$ for each $1\leq i\leq N_2$, and $H_j\nsubseteq P_i$ for each $1\leq i\leq N_1$ and $1\leq j\leq N_2$.

We denote the singular set of $\boldsymbol{C}$ in the embedded sense as $T(\boldsymbol{C})$, which is precisely defined as:

\[ T(\boldsymbol{C}):=\left\{ x \in \mathrm{spt}\|\boldsymbol{C}\|: \mathrm{spt}\|\boldsymbol{C}\| \text{ is not part of a hyperplane near }x\right\}. \]

We denote $T_\tau(\boldsymbol{C})$ as the $\tau$-neighborhood of $T(\boldsymbol{C})$ for $\tau\gt{}0$.

Proposition 4.3.6

Let $n\geq 3$, and $\bar{n}\lt{}n-4+\frac{4}{n}$. Suppose $M_j$ is a sequence of smooth, immersed, two-sided stable minimal hypersurfaces in $B^{n+1}_4(0)$ with $\mathrm{dim}(\mathrm{sing}(M_j)\cap B^{n+1}_4(0))\leq \bar{n}$, such that $M_j$ converge (as varifolds) to $\boldsymbol{C}\lfloor(B^{n+1}_4(0))$ as $j\to \infty$. Then,

\[ \lim_{j\to \infty} \sup_{B^{n+1}_{\frac{1}{2}}(0)\cap M_j}|A_{M_j}|=0. \]

In particular, $\boldsymbol{C}$ is a sum of hyperplanes with multiplicities, $\mathrm{sing} M_j\cap B^{n+1}_{\frac{1}{4}}(0)=\emptyset$, and $M_j\cap B^{n+1}_{\frac{1}{4}}(0)$ converges smoothly to $\boldsymbol{C}$ in $B^{n+1}_{\frac{1}{4}}(0)$ as immersions with $q$ connected components, where $q=\Theta(\|\boldsymbol{C}\|,0)$.

Proof. For each fixed $\tau\gt{}0$, we know that $M_j\cap B^{n+1}_3(0)$ converges smoothly to $\boldsymbol{C}\lfloor (B^{n+1}_3(0)\backslash T_\tau(\boldsymbol{C}))$ as $j\to \infty$ by Proposition 4.3.4. For each $\tau\gt{}0$, by the proof of Proposition 4.3.4, we know

\[ \lim_{j\to \infty} \int_{ B^{n+1}_2(0)\cap M_j\backslash T_\tau(\boldsymbol{C})} |A_{M_j}|^{2\alpha}=0. \]

Now take $k=0$, and let $\phi$ be a nonnegative cutoff function supported in $B^{n+1}_{\frac{3}{2}}(0)$, equal to $1$ in $B^{n+1}_1(0)$, with $|\nabla \phi|\leq 4$. Applying Lemma 4.3.2 together with the Michael–Simon inequality [MS73], we have

\begin{equation} \tag{4.3.11} \begin{aligned} &\int_{ M_j\cap B^{n+1}_1(0)}|A_{M_j}|^{\frac{2\alpha n}{n-2}}\leq C \left( \int_{ M_j\cap B^{n+1}_{\frac{3}{2}}(0)} |A_{M_j}|^{2\alpha} \right)^{\frac{n}{n-2}}\nonumber \\ \leq{}& C \left( \int_{ M_j\cap B^{n+1}_{\frac{3}{2}}(0)} |A_{M_j}|^{2} \right)^{\frac{n\alpha}{n-2}}(\mathcal{H}^n(M_j\cap B^{n+1}_{\frac{3}{2}}(0)))^{\frac{n(1-\alpha)}{n}}, \label{eq:pfSecondUniBound} \end{aligned} \end{equation}

for some $C=C(n)$. Note that the stability condition and Lemma 4.3.3 imply that the right-hand side of (4.3.11) is uniformly bounded. Hence,

\[ \sup_{j\gt{}0}\int_{ M_j\cap B^{n+1}_1(0)} |A_{M_j}|^{\frac{2\alpha n}{n-2}}\lt{}\infty. \]

By Hölder’s inequality, we have

\[ \begin{aligned} &\int_{ M_j\cap T_\tau(\boldsymbol{C}) \cap B^{n+1}_1(0)}|A_{M_j}|^{2\alpha}\\ \leq{} & \left( \int_{ M_j\cap T_\tau(\boldsymbol{C})\cap B^{n+1}_1(0)}|A_{M_j}|^{\frac{2\alpha n}{n-2}} \right)^{\frac{n-2}{n}}\mathcal{H}^n(M_j\cap T_\tau(\boldsymbol{C})\cap B^{n+1}_1(0))^{\frac{2}{n}}.\\ \end{aligned} \]

By a standard covering argument using the monotonicity formula, we know that

\[ \mathcal{H}^n(M_j\cap T_\tau(\boldsymbol{C}) \cap B^{n+1}_2(0))\leq C\tau, \]

for some $C=C(n,\Theta(\|\boldsymbol{C}\|,0))$ for $j$ large enough. Hence, we have

\begin{equation} \tag{4.3.12} \lim_{j\to \infty} \int_{ B^{n+1}_1(0)\cap M_j} |A_{M_j}|^{2\alpha}=\lim_{j\to \infty} \int_{ B^{n+1}_1(0)\cap M_j\cap T_\tau(\boldsymbol{C})} |A_{M_j}|^{2\alpha}\leq C\tau^{\frac{2}{n}}, \label{eq:pfAlphaTau} \end{equation}

for some $C\lt{}\infty$ which is independent of $\tau$. Since the left-hand side of (4.3.12) is independent of $\tau$, and $\tau$ is arbitrary, we obtain

\[ \lim_{j\to \infty} \int_{ B^{n+1}_1(0)\cap M_j} |A_{M_j}|^{2\alpha}=0. \]

Thus, we apply the $\varepsilon$-regularity theorem (Theorem 4.3.1) to conclude that

\[ \lim_{j\to \infty} \sup_{B^{n+1}_{\frac{1}{2}}(0)\cap M_j}|A_{M_j}|=0, \]

which implies that each connected component of $M_j\cap B^{n+1}_{\frac{1}{2}}(0)$ converges to a hyperplane in the varifold sense. Furthermore, by Proposition 4.3.4, for $j$ large enough, $\mathrm{sing}M_j\cap B^{n+1}_{\frac{1}{4}}(0)$ is empty, and $M_j\cap B^{n+1}_{\frac{1}{4}}(0)$ has exactly $q$ connected components, each converging smoothly to a hyperplane in $B^{n+1}_{\frac{1}{4}}(0)$. ◻

We now prove Theorem 4.1.8. By Allard’s compactness theorem (cf. Theorem 3.1.15), we obtain a stationary integral varifold $V$ in $B^{n+1}_4(0)$ such that, up to a subsequence, $|M_j|$ converges to $V$ in the varifold sense. Let $S=\mathrm{sing}\|V\|$ be the singular point set of $V$. We need to analyze the tangent cone $\boldsymbol{C}$ of $V$ at $x_0\in S\cap B^{n+1}_{\frac{1}{2}}(0)$. Indeed, we have the following lemma.

Lemma 4.3.7

For any $\boldsymbol{C} \in \mathrm{VarTan}(V,x_0)$ for $x_0 \in S \cap B^{n+1}_{\frac{1}{2}}(0)$, we can write $\boldsymbol{C}=\boldsymbol{C}'\times \mathbb{R}^{n-p}$ for some $p\geq 7$.

Proof. For any cone $\boldsymbol{C}$, we write $\mathcal{S}(\boldsymbol{C})$ (the spine of $\boldsymbol{C}$) to be the linear subspace containing all $x\in \mathbb{R}^{n+1}$ such that $\boldsymbol{C}$ is invariant under the translation along the line spanned by $x$. For any $x_0 \in S$, we introduce the notion of iterated tangents of $V$ at $x_0$ as follows. We say a collection of cones $\left\{ \boldsymbol{C}_1,\boldsymbol{C}_2,\cdots ,\boldsymbol{C}_N \right\}$ is iterated tangents of $V$ at $x_0$ if $\boldsymbol{C}_1$ is the tangent cone of $V$ at $x_0$, and $\boldsymbol{C}_{j+1}$ is the tangent cone of $\boldsymbol{C}_j$ at $x_j \in \mathrm{sing}\|\boldsymbol{C}_j\|\backslash \mathcal{S}(\boldsymbol{C}_j)$ for $1\leq j\leq N-1$. Moreover, we can choose iterated tangents satisfying the following properties:

Each $\boldsymbol{C}_j$ is not smoothly immersed (i.e., $\mathrm{sing}\|\boldsymbol{C}_j\|\neq \emptyset$).
$\mathrm{dim}(\mathcal{S}(\boldsymbol{C}_{j+1}))\gt{}\mathrm{dim}(\mathcal{S}(\boldsymbol{C}_{j}))$ for each $j=1,2,\cdots ,N-1$.
$\boldsymbol{C}_N=\boldsymbol{C}'\times \mathbb{R}^{\mathrm{dim}(\mathcal{S}(\boldsymbol{C}_N))}$ where $\boldsymbol{C}'\backslash \left\{ 0 \right\}$ is a smooth immersed cone after a suitable rotation in $\mathbb{R}^{n+1}$.
For each $1\leq j\leq N$, we can find a sequence of points $\left\{ y_k \right\}$ with $y_k\to x_0$, a sequence of positive real numbers $\left\{ r_k \right\}$ with $r_k\to 0^+$ as $k\to \infty$, such that $\eta_{y_k,r_k}(M_k)$ converges to $\boldsymbol{C}_j$ in the sense of varifolds and the convergence is smooth away from the singular set of $\boldsymbol{C}_j$ by Proposition 4.3.4 and Proposition 4.3.6.

In particular, the fourth condition implies that the smooth immersed part of $\boldsymbol{C}_j$ is stable, and the second condition implies $N$ is a finite number.

The first three conditions are immediate from properties of tangent cones. The only nontrivial part is the last condition, which can be proved by induction on $j$. Suppose we have found $y_k,r_k$ such that $\eta_{y_k,r_k}(M_k)$ converges to $\boldsymbol{C}_j$ in the sense of varifolds. Then, by the choice of $\boldsymbol{C}_{j+1}$, we know there exists $\rho_k$ such that $\eta_{x_j,\rho_k}(\boldsymbol{C}_j)$ converges to $\boldsymbol{C}_{j+1}$ in the sense of varifolds. Thus, with $z_k=y_k+r_k x_j$ and $s_k=r_k \rho_k$, we have $\eta_{z_k,s_k}(M_k)$ converging to $\boldsymbol{C}_{j+1}$ in the varifold sense.

Now, let us determine the dimension of $\boldsymbol{C}'$. Note that $\boldsymbol{C}_N$ cannot be a hyperplane by the first condition.

If the dimension of $\boldsymbol{C}'$ is one, then $\boldsymbol{C}_N$ is the sum of distinct half-hyperplanes with multiplicity. But by Proposition 4.3.6, we know $\boldsymbol{C}_N$ can only be a sum of hyperplanes with multiplicity, which contradicts the first condition.

Therefore, we know $\boldsymbol{C}'$ has dimension at least two. But the fourth condition implies that $\boldsymbol{C}'$ is a smooth immersed stable cone away from $\left\{ 0 \right\}$, and hence, $\boldsymbol{C}'$ has dimension at least $7$.

Hence, by the second condition, we obtain $\mathrm{dim}(\boldsymbol{C})\geq n-7$ for any $\boldsymbol{C}\in \mathrm{VarTan}(V,x_0)$, and the lemma follows. ◻

Now, we are ready to finish the proof of Theorem 4.1.8.

Theorem 4.3.8

Let $\mathcal{V}$ be the collection of all the limit varifolds defined in Theorem 4.1.8. Then,

\[ \operatorname{dim}(\mathrm{sing}(\|V\|\cap B_1))\leq n-7. \]

In particular, if $n=7$, then $\mathrm{sing}(\|V\|\cap B_1)$ is discrete.

Proof. We denote $F^l=\{ V \in \mathcal{V}: \mathcal{H}^l(\mathrm{sing}\cap B_1)\gt{}0 \}$.

Proposition 4.3.9

For each $V \in F^l$, there exists $\boldsymbol{C} \in \mathrm{VarTan}(V,x)\cap F^l$ for $\mathcal{H}^l-$a.e. $x \in \mathrm{sing}(\|V\|)\cap B_1$.

Proof. Recall that we actually have for $\mathcal{H}^l$-a.e. $x \in \mathrm{sing}(\|V\|)\cap B_1$, we have

\[ \limsup_{r \to 0} \frac{\mathcal{H}^l_\infty(\text{sing}\|V\|\cap B_r(x))}{\omega_n r^l}\gt{}0. \]

We choose $r_i \to 0$ such that

\[ \lim_{i \to \infty} \frac{\mathcal{H}^l_\infty(\text{sing}\|V\|\cap B_{r_i}(x))}{\omega_n r_i^l}\gt{}0. \]

By taking a subsequence, we can assume $(\eta_{x,r_i})_{\#}V$ converges to $\boldsymbol{C} \in \mathrm{VarTan}(V,x)$.

\[ \sum_{j=1}^\infty s_j^l \lt{} \varepsilon. \]

\[ \frac{\mathcal{H}^l_\infty(\mathrm{sing}\,\|V\|\cap B_{r_i}(x))}{r_i^l}\lt{}\varepsilon \]

for $i$ large enough, which is a contradiction. ◻

Now we apply the above proposition iteratively to obtain a sequence of varifolds $\{ V_k \}_{k=0}^{K}$ such that:

$V_0=V$.
$V_{k+1} \in \mathrm{VarTan}(V_k,x_k)$ for some $x_k \in \mathrm{sing}(\|V_k\|)\cap B_1 \backslash \mathcal{S}(V_k)$.
$\operatorname{dim}(\mathcal{S}(V_{k+1}))\gt{}\operatorname{dim}(\mathcal{S}(V_k))$ for each $0\leq k \leq K-1$.
$\mathcal{H}^l(\mathrm{sing}(\|V_k\|)\cap B_1)\gt{}0$ for each $0\leq k \leq K$.
$V_K=\boldsymbol{C}\times \mathbb{R}^{m}$ where $\boldsymbol{C}\backslash \{0\}$ is a smooth immersed cone for some $m\geq 0$.

In particular, the last two conditions imply $m=l$. By Lemma 4.3.7, we know $l \leq n-7$.

In the case $n=7$, we have

\[ \mathcal{H}^\alpha(\mathrm{sing}\,\|V_K\|\cap B_1)=0 \]

for any $\alpha\gt{}0$ by the above result.

Sub-minimal Sets and Sub-calibrations

Mon, 29 Jun 2026 00:00:00 +0000

Let $\Omega\subset\mathbb{R}^n$ be an open set. For a set $E$ with smooth boundary, the perimeter $P(E,A)$ in a smooth bounded open set $A\subset\Omega$ is the $(n-1)$-dimensional Hausdorff measure of $\partial E\cap A$.

Definition 2.3.1

A set $E$ with smooth boundary is sub-minimal in $\Omega$ if for every smooth bounded open set $A\subset\Omega$ and every $F\subset E$ with $E\setminus F\subset\subset A$,

\[ P(E,A)\leq P(F,A). \]

Proposition 2.3.2

If $E$ and $E^c=\Omega\setminus E$ are both sub-minimal in $\Omega$, then $E$ is minimal in $\Omega$.

Proof. Let $A\subset\Omega$ be a smooth bounded open set, and let $F$ satisfy $E\triangle F\subset\subset A$. Define $F'=E\cap F\subset E$ and $F''=(E\cup F)^c\subset E^c$. Then $E\setminus F'\subset\subset A$ and $E^c\setminus F''\subset\subset A$. By sub-minimality:

\[ P(E,A)\leq P(F',A),\quad P(E^c,A)\leq P(F'',A). \]

Since $P(E^c,A)=P(E,A)$ and $P(F'',A)=P(E\cup F,A)$, we have

\[ 2P(E,A)\leq P(E\cap F,A)+P(E\cup F,A). \]

Using the identity

\[ P(E\cap F,A)+P(E\cup F,A)\leq P(E,A)+P(F,A), \]

we conclude $P(E,A)\leq P(F,A)$, so $E$ is minimal. ◻

Definition 2.3.3

Let $E$ have smooth boundary. A $C^1$ vector field $\xi$ on $\Omega$ is a sub-calibration of $E$ if:

$\xi(x)=\nu_E(x)$ (exterior unit normal) for all $x\in\partial E$;
$\operatorname{div}\xi(x)\leq0$ for all $x\in E$;
$\lvert\xi(x)\rvert\leq1$ for all $x\in\Omega$.

Theorem 2.3.4

If $E$ admits a sub-calibration $\xi$, then $E$ is sub-minimal.

Proof. Let $A$ be a bounded open set, and let $F\subset E$ with $E\setminus F\subset\subset A$. Choose $\eta_j\in C_c^1(A)$ with $\eta_j=1$ on $E\setminus F$, $0\leq\eta_j\leq1$, and $\bigcup_j\{x:\eta_j(x)=1\}=A$. Let $\xi_j=\eta_j\xi$. Then

\[ \int_{E\cap A}\operatorname{div}\xi_j - \int_{F\cap A}\operatorname{div}\xi_j = \int_{E\setminus F}\operatorname{div}\xi \leq 0, \]

\[ \int_{E\cap A}\operatorname{div}\xi_j \leq \int_{F\cap A}\operatorname{div}\xi_j \leq P(F,A). \]

By the divergence theorem:

\[ \int_{E\cap A}\operatorname{div}\xi_j = \int_{\partial E\cap A}\langle\xi_j,\nu_E\rangle d\mathcal{H}^{n-1} = \int_{\partial E\cap A}\eta_j d\mathcal{H}^{n-1} \geq \mathcal{H}^{n-1}\bigl(\partial E\cap\{\eta_j=1\}\bigr). \]

Taking $j\to\infty$ yields $P(E,A)\leq P(F,A)$, so $E$ is sub-minimal. ◻

Let $n=2m$. The Simons cone is

\[ \mathbf C=\bigl\{(x_1,\dots,x_m,y_1,\dots,y_m)\in\mathbb{R}^{2m}: x_1^2+\dots+x_m^2=y_1^2+\dots+y_m^2\bigr\}. \]

Define

\[ \mathcal{C}=\bigl\{(x,y)\in\mathbb{R}^m\times\mathbb{R}^m: |x|\leq|y|\bigr\},\quad f(x,y)=\frac{1}{4}\bigl(|x|^4-|y|^4\bigr). \]

Define the vector field

\[ \xi=\frac{Df}{|Df|}. \]

Proposition 2.3.5

$\xi$ is a sub-calibration for $E$ in $\mathbb{R}^{2m}\setminus\{0\}$, where $E:=\{(x,y):|x|\leq |y|\}$, and $-\xi$ is a sub-calibration for $E^c$ in $\mathbb{R}^{2m}\setminus\{0\}$.

Proof.

$\lvert\xi\rvert=1$ everywhere, and $\xi$ is the exterior normal to $E$ on $\partial E$. We compute

\[ Df=(|x|^2 x,\,-|y|^2 y), \qquad |Df|^2 = |x|^6 + |y|^6. \]

Hence

\[ \Delta f = (m+2)|x|^{2} - (m+2)|y|^{2},\quad D|Df|^{2}= 6\bigl(|x|^{4} x,\, |y|^{4} y\bigr). \]

Thus

\[ \begin{aligned} |Df|^3\mathrm{div}\xi ={}& |Df|^2 \Delta f - \frac{1}{2}\langle D|Df|^2, Df\rangle\\ ={}& \bigl((m+2)|x|^{2} - (m+2)|y|^{2}\bigr)(|x|^6 + |y|^6) - 3\bigl(|x|^{8} + |y|^{8}\bigr)\\ ={}& (m-1)|x|^8 - (m-1)|y|^8 + (m+2)|x|^{2}|y|^{6} - (m+2)|x|^{6}|y|^{2}\\ ={}& (m-1)(|x|^4-|y|^4)(|x|^4+|y|^4) + (m+2)|x|^{2}|y|^{2}(|y|^4 - |x|^4)\\ ={}& (|x|^4 - |y|^4)\bigl((m-1)(|x|^4 + |y|^4) - (m+2)|x|^2 |y|^2\bigr). \end{aligned} \]

Note that

\[ (m-1)a^{2} - (m+2)ab + (m-1)b^{2} \]

is nonnegative if and only if

\[ \mathrm{det}\,\begin{pmatrix} m-1 & -\frac{m+2}{2}\\ -\frac{m+2}{2} & m-1 \end{pmatrix} = (m-1)^2 - \frac{(m+2)^2}{4} = \frac{3m(m-4)}{4} \geq 0. \]

◻

Theorem 2.3.6

The Simons cone $\mathbf C$ is area-minimizing in $\mathbb{R}^8$.

Proof. By the preceding theorem, $E$ and $E^c$ are sub-minimal in $\mathbb{R}^8\setminus\{0\}$. Since the origin has codimension $\gt{}1$, the perimeter is unchanged, and hence $E$ and $E^c$ are sub-minimal in $\mathbb{R}^8$. By the preceding proposition, $E$ is minimal, so its boundary $\mathbf C$ is area-minimizing. ◻

One may consider the gradient of the function

\[ \frac{1}{4}\left( q^{2}|x|^4- p^{2}|y|^4 \right) \]

in the case of the minimizing cone $\mathbf C_{p,q}$.

4. The first computational phase: explicit formulas

Thu, 27 Aug 2026 00:00:00 +0000

The purpose of this phase was to determine a basic form for the test function, and especially to decide which choices of the background $S$ and detector $P$ in (2.5) were most effective. We focused first on $\ell=2$, asking which functions $G(\nu_1,\nu_2)$ could survive numerical screening.

4.1 The first numerical candidates

Put $S=1-\nu_1^2-\nu_2^2$. The first ten tested candidates are collected below. Most belong to the $\ell=2$ search; the final two are the immediate higher-rank product variants tested in the same phase.

Candidate $G=g^2$	Numerical screening
$G=1-\nu_1^2-\nu_2^2+k(\cos\vartheta-\nu_1)^2$	✓ Passed
$G=S+k\nu_1^2\nu_2^2$	✕ Failed
$G=S^2+k(\nu_1-\nu_2)^2(\nu_1+\nu_2-1)^2$	✕ Failed
$G=\bigl[S^2+k(\nu_1-\nu_2)^2(\nu_1+\nu_2-1)^2\bigr]^{1/2}$	✕ Failed
$G=\left[\prod_{\pm}\left(1-\frac{\sqrt{3}}{2}\nu_1\pm\frac{1}{2}\nu_2\right)(1+\nu_2)\right]^{2/3}$	✓ Passed
$G=\left[S^{3/2}+k\prod_{\pm}\left(1-\frac{\sqrt{3}}{2}\nu_1\pm\frac{1}{2}\nu_2\right)(1+\nu_2)\right]^{2/3}$	✕ Failed
$G=S+k(1-\nu_1^2)(1-\nu_2^2)$	✕ Failed
$G=\bigl[S^2+k(1-\nu_1^2)(1-\nu_2^2)\bigr]^{1/2}$	✕ Failed
$G=\bigl[(1-\nu_1)(1-\nu_2)(1-\nu_3)\bigr]^{2/3}$	✓ Passed
$G=\bigl[(1-\nu_1^2)(1-\nu_2^2)(1-\nu_3^2)(1-\nu_4^2)\bigr]^{1/4}$	✕ Failed

Table 4.1. The first ten candidate formulas and their numerical screening outcomes.

These numerical screenings ruled out many natural alternatives and led us to focus on candidates of the form $G=S+kP$, where $S=1-\nu_1^2-\nu_2^2$ and $P$ vanishes at the prescribed directions. In particular, several simple choices, including $P=\nu_1^2\nu_2^2$ and $P=(1-\nu_1^2)(1-\nu_2^2)$, failed the numerical screening. This made clear that $P$ had to be designed more carefully in order to obtain a viable candidate.

4.2 Searching over the detector $P$

The next round of experiments focus on $\ell=3, 4$. Thus the background was

\[ S=1-\nu_1^2-\nu_2^2-\cdots -\nu_\ell^2, \qquad G=S+kP. \]

We varied the structure of $P$, testing products of affine pole factors, fractional powers, repeated factors, and tilted factors.

Candidate $P$	Numerical screening
Rank-three tests: $S=1-\nu_1^2-\nu_2^2-\nu_3^2,\quad t\in [0,2\pi]$
$P=(1-\nu_1)(1-\nu_2)$	✓ Passed
$P=\bigl[(1-\nu_1)(1-\nu_2)(1-\nu_1/\sqrt{2}-\nu_3/\sqrt{2})\bigr]^{2/3}$	✕ Failed^*
$P=\bigl[(1-\nu_1)(1-\nu_2)(1-\nu_1\cos t-\nu_3\sin t)\bigr]^{2/3}$	✕ Failed^*
$P=\bigl[(1-\nu_1)^2(1-\nu_2)\bigr]^{2/3}$	✕ Failed^*
$P=\bigl[(1-\nu_1)(1-\nu_2)(1-\nu_3)(1-\nu_1\cos t-\nu_3\sin t)\bigr]^{1/2}$	✕ Failed^*
$P=\bigl[(1-\nu_1)^2(1-\nu_2)(1-\nu_3)\bigr]^{1/2}$	✕ Failed^*
$P=\bigl[(1-\nu_1)^2(1-\nu_2)(1-\nu_2\cos t-\nu_3\sin t)\bigr]^{1/2}$	✕ Failed^*
$P=\left[\prod_{i=1}^2(1-\nu_i)(1-\nu_i\cos t-\nu_3\sin t)\right]^{1/2}$	✕ Failed^*
$P=\bigl[(1-\nu_1^2)(1-\nu_2^2)(1-\nu_3^2)\bigr]^{1/3}$	✕ Failed^*
Additional $n=6$ tests: $S=1-\nu_1^2-\nu_2^2-\nu_3^2-\nu_4^2$
$P=\frac{1}{4}\bigl[(1-\nu_1)(1-\nu_2)(1-\nu_3)(1+\nu_3)\bigr]^2$	✕ Failed
$P=(1-\nu_1)(1-\nu_2)(1-\nu_3)$	✕ Failed
$P=(1-\nu_1)(1-\nu_2)$	✕ Failed

Table 4.2. Further choices of $P$ and their numerical screening status.

Interpreting the starred failures.

Every formula marked “Failed$^{*}$” failed for the same localized reason: negative values were detected only in thin neighborhoods of one or more prescribed directions. On the sampled regions bounded away from the zeros, the numerical margin remained positive. As $k$ decreased, the bad neighborhoods became smaller and were therefore easy to miss at fixed numerical resolution. An unstarred “Failed” means that negative values were already visible in the retained general scan, rather than only in a zero neighborhood.

Thus the experiments were almost entirely negative, but the location of the failure was encouraging: away from the zeros, the starred candidates retained a positive margin. This suggested that the global product structure might still be useful and that only the local vanishing model had to be repaired. Even a sheeting theorem for three arbitrary prescribed directions would already have been a worthwhile result, but at this stage no choice of $P$ passed all the relevant tests.

A second pattern identified the exponent itself. Every starred candidate shared the balanced power structure

\[ \class{key-formula-math}{P=\prod_{j=1}^m L_j^{2/m}, \qquad L_j=1-\ip{\nu}{p_j},} \]

with repeated or tilted factors allowed.

Observation 4.1

Beyond the formulas listed in the table, we numerically tested many products with other exponents and several non-power modifications. These additional candidates developed negative values away from the zero set. By contrast, the observed failures of the balanced $2/m$-power products were confined to shrinking neighborhoods of the zeros.

The data now pointed both to a candidate exponent and to a rank restriction. While the balanced products behaved consistently in ranks two and three, the rank-four experiments were markedly less stable: new negative directions appeared away from the zeros, and neither decreasing $k$ nor adding more factors removed them. We therefore restricted the next stage of the search to $\ell\leq3$ and replaced the open-ended search for new formulas by an analysis of the necessary conditions on $P$. In particular, we examined the regime $k\downarrow0$ to determine whether the exponent $2/m$ was indeed sufficient on every compact set away from the zeros, to understand why the bad neighborhoods shrank, and to isolate the additional local condition that would eventually be needed at a zero. This change of viewpoint led directly to the next section.

Bellettini’s Sheeting Theorem and Schoen–Simon Regularity

Mon, 29 Jun 2026 00:00:00 +0000

Theorem 4.4.1

Suppose $M$ is a properly immersed, two-sided, stable minimal hypersurface in $B^{n+1}_4(0)$ with $\mathcal{H}^{n-2}(\mathrm{sing}(M))\lt{}\infty$ and

\[ \mathcal{H}^n(M\cap B^{n+1}_4(0))\leq \Lambda . \]

Then there exists $\varepsilon=\varepsilon(n,\Lambda)\gt{}0$ such that if

\[ \int_{M\cap B^{n+1}_4(0)} \text{dist}(x,P)^2 d\mathcal{H}^n \leq \varepsilon \]

for $P=\{ x_{n+1}=0 \}$, then $\pi:M \cap B^n_1(0)\times \mathbb{R}\backslash \pi^{-1}(\Sigma)\to B^n_1(0)\backslash \Sigma$ is a smooth projection map, where $\Sigma$ is the projection of $\text{sing}(M)$ to $P$ in $B^n_1(0)$.

In particular, if $\mathcal{H}^{n-2}(\text{sing}(M))=0$, then $\Sigma=\emptyset$ and $M\cap B^n_1(0)\times \mathbb{R}$ is a union of minimal graphs over $B^n_1(0)$.

Theorem 4.4.2

Suppose $M$ is a properly immersed, two-sided, stable minimal hypersurface in $B^{n+1}_4(0)$ with $\mathcal{H}^{n-2}(\mathrm{sing}(M))\lt{}\infty$ and

\[ \mathcal{H}^n(M\cap B^{n+1}_4(0))\leq \Lambda . \]

Then, there exists $\varepsilon=\varepsilon(n,\Lambda)\gt{}0$ such that if

\[ \int_{M\cap B^{n+1}_4(0)} |x_{n+1}|^2 d\mathcal{H}^n \leq \varepsilon, \]

then

\[ g(x)\leq C \left( \int_{M\cap B_{2}(0)} g^{2}(x) \right)^{\frac{4}{4+n}}. \]

Here, $g(x)=\sqrt{1-(\nu \cdot e_{n+1})^{2}}$ where $\nu$ is the unit normal vector of $M$.

Geometric meaning of $g$.

Definition 4.4.3

The tilt excess of $M$ in $B^{n+1}_r(0)$ with respect to $P=\{ x_{n+1}=0 \}$ is

\[ E_M(r,P):=r^{-n}\int_{M\cap B^{n+1}_r(0)} |T_x M - P|^{2} d\mathcal{H}^n. \]

For the codimensional one case, one can use $|\nu-e_{n+1}||\nu+e_{n+1}|$ as the distance between $T_x M$ and $P$. We have

\[ |T_xM-P|^{2}\simeq |\nu-e_{n+1}|^{2}|\nu+e_{n+1}|^{2}=4(1-(\nu \cdot e_{n+1})^{2})=4g^{2}. \]

This agrees with the standard definition of tilt excess up to a constant multiple.

Lemma 4.4.4

For any $k\in [0,\frac{1}{2n}]$, we have

\[ \frac{1}{2n}\int_{ \{ g\gt{}k \}} |\nabla g|^{2} \left( 1-\frac{k}{g} \right) \phi^{2} \leq \int_{\{ g\gt{}k \} } (g-k) ^{2}|\nabla \phi|^{2}. \]

for any Lipschitz function $\phi$ supported in $B^{n+1}_{\frac{3}{2}}(0)$.

Proof. Choose $(g-k)^+ \phi$ as a test function in the stability inequality, together with

\[ g\Delta g = - \frac{|\nabla g|^{2}}{1-g^{2}}+|A|^{2}(1-g^{2}) \]

and

\[ \frac{|\nabla g|^{2}}{1-g^{2}}\leq \frac{n-1}{n}|A|^{2}, \]

we can finish the proof for this lemma for the $\phi$ with compact support in the regular part of $M$. The general case can be obtained by a standard cut-off argument near the singular set of $M$.

The proof for $\varepsilon$-regularity theorem for the tilt is similar to the proof of the previous $\varepsilon$-regularity theorem for $|A|$, as we only need to repeat the de Giorgi iteration process.

The proof of the sheeting theorem is as follows.

Note that by the 1st variation formula, (by choosing $X = \phi^{2}x_{n+1}e_{n+1}$) we have

\[ \int_{ } g^{2} \phi^{2} \leq 4 \int_{ } |x_{n+1}|^{2} |\nabla \phi|^{2}. \]

\[ \int_{ B_2} g^{2} \leq C \int_{ B_4} |x_{n+1}|^{2} \leq C \varepsilon \]

Then, given any $\delta\gt{}0$, we can choose $\varepsilon$ small enough such that $g\lt{}\delta$ in $B_1^n \times \mathbb{R}\cap \text{reg}(M)$ by the $\varepsilon$-regularity theorem for the tilt. In particular, each small regular region of $M\cap B_1^n \times \mathbb{R}$ can be written as a graph over a domain in $P$ with small gradient. Then, by the connectedness of $M\cap B_1^n \times \mathbb{R}$ away from the singular set, we know $M\cap B_1^n \times \mathbb{R}\backslash \pi^{-1}(\Sigma)$ can be written as a graph of smooth multiple valued function (locally, it is the union of smooth graphs) over $B_1^n \backslash \Sigma$. This can be viewed as a covering map from $M\cap B_1^n \times \mathbb{R}\backslash \pi^{-1}(\Sigma)$ to $B_1^n \backslash \Sigma$.

In addition, if $\mathcal{H}^{n-2}(\text{sing}(M))=0$, then $B_1^n \backslash \Sigma$ is simply connected. So the covering map is trivial, and hence $M\cap B_1^n \times \mathbb{R}$ is a union of minimal graphs over $B_1^n(0)\backslash \Sigma$. Again by the removable singularity theorem, each graph can be extended to a smooth minimal graph over $B_1^n(0)$. ◻

Theorem 4.4.5

Let $\{M_k\}$ be a sequence of embedded, stable, orientable minimal hypersurfaces in $B_2^{n+1}(0)$ with the following properties:

$0\in \bar{M}_k$ for each $k$.
$\mathcal{H}^{n-2}(\mathrm{sing}M_k)=0$ for each $k$.
$\|M_k\|(B_2^{n+1}(0))\leq \Lambda$ for some constant $\Lambda\gt{}0$ independent of $k$.

Proof. By Allard’s compactness theorem, up to a subsequence we have $|M_i|\to V$ in the varifold sense.

Let $S=\mathrm{sing}\|V\|$ be the embedded singular point set of $V$. By Federer’s dimension-reduction argument, it suffices to prove the following lemma.

Lemma 4.4.6

For any $\boldsymbol{C} \in \mathrm{VarTan}(V,x_0)$ for $x_0 \in S \cap B^{n+1}_{\frac{1}{2}}(0)$, we can write $\boldsymbol{C}=\boldsymbol{C}'\times \mathbb{R}^{n-p}$ for some $p\geq 7$.

The proof of this lemma is similar to that of Lemma 4.3.7. We construct the iterated tangents of $V$ at $x_0 \in S$ as $\{ \boldsymbol{C}_1,\cdots , \boldsymbol{C}_N \}$ with the following:

$x_j \in \mathrm{sing}\|\boldsymbol{C}_j\|\backslash \mathcal{S}(\boldsymbol{C}_j)$ for each $1\leq j\leq N-1$ and $\boldsymbol{C}_{j+1} \in \mathrm{VarTan}(\boldsymbol{C}_j,x_j)$.
Each $\boldsymbol{C}_j$ is not smoothly embedded (i.e., $\mathrm{sing}\|\boldsymbol{C}_j\|\neq \emptyset$).
$\mathrm{dim}(\mathcal{S}(\boldsymbol{C}_{j+1}))\gt{}\mathrm{dim}(\mathcal{S}(\boldsymbol{C}_{j}))$ for each $j=1,2,\cdots ,N-1$.
$\boldsymbol{C}_N=\boldsymbol{C}'\times \mathbb{R}^{\mathrm{dim}(\mathcal{S}(\boldsymbol{C}_N))}$ where $\boldsymbol{C}'\backslash \left\{ 0 \right\}$ is a smooth embedded cone after a suitable rotation in $\mathbb{R}^{n+1}$.
For each $1\leq j\leq N$, we can find a sequence of points $\left\{ y_k \right\}$ with $y_k\to x_0$, a sequence of positive real numbers $\left\{ r_k \right\}$ with $r_k\to 0^+$ as $k\to \infty$, such that $\eta_{y_k,r_k}(M_k)$ converges to $\boldsymbol{C}_j$ in the sense of varifolds and the convergence is smooth away from the singular set of $\boldsymbol{C}_j$ by sheeting theorem.

We need to show that $\boldsymbol{C}'$ has dimension at least $7$. If the dimension of $\boldsymbol{C}'$ is one, then $\boldsymbol{C}_N$ is the sum of distinct half-hyperplanes with multiplicity.

For simplicity, we denote

\[ \boldsymbol{C}_N=\sum_{i=1}^{N_1} q_i \{ (\cos \theta_i r, \sin \theta_i r, y): r \geq 0, y \in \mathbb{R}^{n-1} \}. \]

We denote $\hat{M}_k=\eta_{y_k,r_k}(M_k)$. By the sheeting theorem, for any $\tau\gt{}0$ and all large $k$, the set

\[ \hat{M}_k \cap B^{n+1}_2(0)\backslash T_\tau(\boldsymbol{C}_N) \]

decomposes into $q=\sum_{i=1}^{N_1} q_i$ connected components. Each component is a smooth graph over the corresponding half-hyperplane in $\boldsymbol{C}_N$.

Now, for $\mathcal{H}^{n-1}$-a.e. point $y \in \mathbb{R}^{n-1}\cap B_1^{n-1}$, Sard’s theorem and $\mathcal{H}^{n-2}(\mathrm{sing}\,\hat{M}_k)=0$ imply that $\hat{M}_k \cap B_1^{2} \times \{ y \}$ consists of $q$ embedded curves for large $k$.

Choose two pieces $N_1,N_2 \subset \hat{M}_k\backslash T_\tau(\boldsymbol{C}_N)$ such that $N_i$ is a graph over a domain in $\{ (\cos \theta_i r, \sin \theta_i r, y): r \geq \tau, y \in \mathbb{R}^{n-1} \}$ and $N_1,N_2$ are connected by a curve $\gamma$ as above.

Note that $\left\vert \nu(\gamma \cap \{ r=\tau \}\cap N_1)-\nu(\gamma \cap \{ r=\tau \}\cap N_2) \right\vert \geq \frac{1}{2}|\sin(\theta_1 - \theta_2)|$ for $k$ large enough. So the integral

\[ \int_{ \gamma \cap \{ r\lt{}\tau \}} |A|\geq \frac{1}{2}|\sin(\theta_1 - \theta_2)|. \]

Now, we integrate over all such curves $\gamma$ for $\mathcal{H}^{n-1}$-a.e. point $y \in \mathbb{R}^{n-1}\cap B_1^{n-1}$, we have

\[ \int_{ \hat{M}_k \cap B_\tau^{2} \times B_1^{n-1}} |A| \geq C \min_{i\neq j}|\theta_i-\theta_j|. \]

But on the other hand, using Cauchy-Schwarz inequality, we have

\[ \begin{aligned} \int_{ \hat{M}_k \cap B_\tau^{2} \times B_1^{n-1}} |A| &\leq \left( \mathcal{H}^n(\hat{M}_k \cap B_\tau^{2} \times B_1^{n-1}) \right)^{\frac{1}{2}} \left( \int_{ \hat{M}_k \cap B_\tau^{2} \times B_1^{n-1}} |A|^{2} \right)^{\frac{1}{2}} \nonumber \\ &\leq C \sqrt{\tau} \end{aligned} \]

for some $C=C(n,\Lambda)$ independent of $\tau$ and $k$ by the monotonicity formula and the stability inequality. Since $\tau$ is arbitrary, this is a contradiction.

Hence, we know $\boldsymbol{C}'$ has dimension at least $2$. But the fifth condition implies that $\boldsymbol{C}'$ is a smooth embedded stable cone away from $\{0\}$, and hence $\boldsymbol{C}'$ has dimension at least $7$. ◻

Proof of Bernstein Theorem

Mon, 29 Jun 2026 00:00:00 +0000

Now, we are ready to prove the Bernstein Theorem up to the dimension $n\leq 7$.

Theorem 3.4.1

Suppose $u$ is a solution of the minimal surface equation on $\mathbb{R}^n$ and $n\leq 7$. Then, $u$ is an affine function.

More analysis on the stationary cones

Recall that the Jacobi operator is defined as

\[ L(\varphi) = \Delta \varphi + |A|^{2}\varphi \]

for any smooth function $\varphi$ on the regular part of $M$. If $M^n\subset\mathbb{R}^{n+1}$ is a stationary cone with isolated singular point $0$, then we can rewrite the Jacobi operator as

\[ L(\varphi)=\frac{\partial ^{2}\varphi}{\partial r^2}+\frac{n-1}{r}\frac{\partial \varphi}{\partial r}+\frac{1}{r^{2}}\left( \Delta^\Sigma \varphi + |A_\Sigma|^{2}\varphi \right) \]

where $\Sigma=M \cap \mathbb{S}^{n}$. This is called the link of the stationary cone, which is a minimal hypersurface in $\mathbb{S}^{n}$.

So it is important to study the operator $\Delta^\Sigma + |A_\Sigma|^{2}$ on the link $\Sigma$. Suppose $\Sigma$ is not totally geodesic, we define $\lambda_1(\Sigma)$ to be the first eigenvalue of the operator $-\Delta^\Sigma - |A_\Sigma|^{2}$ by

\[ \lambda_1(\Sigma)=\inf_{\varphi \in H^1(\Sigma), \varphi \neq 0} \frac{\int_\Sigma |\nabla \varphi|^2 - |A_\Sigma|^2 \varphi^2 d\Sigma}{\int_\Sigma \varphi^2 d\Sigma} \]

Theorem 3.4.2

We have $\lambda_1(\Sigma)\leq -(n-1)$.

Proof. We use the following Simons inequality for the link $\Sigma$:

\[ |A_\Sigma|\Delta^\Sigma|A_\Sigma|+|A_\Sigma|^4\geq \frac{2}{n-1}|\nabla |A_\Sigma||^2+(n-1)|A_\Sigma|^2. \]

Now, we directly integrate the inequality over $\Sigma$ to get

\[ \int_{\Sigma} |\nabla |A_\Sigma||^2 - |A_\Sigma|^4 \leq -(n-1)\int_{\Sigma} |A_\Sigma|^2. \]

Using $|A_\Sigma|$ as a test function in the Rayleigh quotient gives

\[ \lambda_1(\Sigma)\leq \frac{\int_\Sigma |\nabla |A_\Sigma||^2-|A_\Sigma|^4\,d\Sigma} {\int_\Sigma |A_\Sigma|^2\,d\Sigma} \leq -(n-1). \]

◻

Proposition 3.4.3

Suppose $\varphi$ is a smooth function on $\Sigma$ achieving the infimum in the definition of $\lambda_1(\Sigma)$. If $u$ is a positive Jacobi field on $M$, then if we define

\[ V(r):=\int_{ \Sigma} \varphi(x) u(r x) d\Sigma(x) \]

Then

\[ (V(r)r^{\kappa_n})'\leq 0 \]

for $\kappa_n:=\frac{n-2}{2}-\sqrt{\frac{(n-2)^{2}}{4}-(n-1)}$.

Proof. We compute

\[ \begin{aligned} V''(r) &= \int_{\Sigma} \varphi(x) \frac{\partial^2}{\partial r^2}u(r x) \, d\Sigma(x) \\ &= -\frac{n-1}{r} V'(r) - \frac{1}{r^2} \int_{\Sigma} \varphi(x) \big( \Delta^\Sigma u(r x) + |A_\Sigma|^2 u(r x) \big) \, d\Sigma(x) \\ &= -\frac{n-1}{r} V'(r) - \frac{1}{r^2} \int_{\Sigma} u(r x) \big( \Delta^\Sigma \varphi(x) + |A_\Sigma|^2 \varphi(x) \big) \, d\Sigma(x) \\ &\leq -\frac{n-1}{r} V'(r) - \frac{n-1}{r^2} V(r) \end{aligned} \]

where in the last step we used that $\varphi$ is an eigenfunction of the operator $-\Delta^\Sigma - |A_\Sigma|^2$ with eigenvalue $\lambda_1(\Sigma) \leq -(n-1)$, and that $u\gt{}0$. The proof is finished by analyzing the resulting ODE inequality. ◻

We consider $W(t)=V(t^{-1/\beta})t^{\gamma/\beta}$.

Then, we can choose $\beta$ and $\gamma$ such that $W''(t)\leq 0$. To see this, we compute

\[ \begin{aligned} W''(t) &= \frac{1}{\beta^2} t^{\frac{\gamma}{\beta}-2} \left( V''(t^{-1/\beta})t^{-2/\beta} + V'(t^{-1/\beta})t^{-1/\beta}((1+\beta)-2\gamma) + V(t^{-1/\beta})\gamma(\gamma-\beta) \right) \\ \end{aligned} \]

So we have

\[ 1+\beta-2\gamma = \gamma(\gamma-\beta)=n-1 \]

The choice of $\gamma = -\kappa_n$ and $\beta = 2\sqrt{\frac{(n-2)^{2}}{4}-(n-1)}$ satisfies the above equations, and hence $W''(t)\leq 0$. Since $t=r^{-\beta}$ and $W(t)=V(r)r^{\kappa_n}$, this concavity gives the desired monotonicity of $V(r)r^{\kappa_n}$ in the $r$ variable.

We are ready to prove the Bernstein Theorem. We need to study the blow-down limit of the graph of $u$. This is due to De Giorgi [DG65].

Theorem 3.4.4

Let $u$ solve the minimal surface equation on $\mathbb{R}^n$ for $n\leq 7$. Then the blow-down limit of the graph of $u$ is a density-one hyperplane in $\mathbb{R}^{n+1}$.

Proof. Suppose $M$ is the graph of $u$. Then it is a minimal hypersurface in $\mathbb{R}^{n+1}$, and it is minimizing area in $\mathbb{R}^{n+1}$.

Let $P$ be the region $\{ (x,y) \in \mathbb{R}^n \times \mathbb{R}: y\lt{}u(x) \}$. By the previous result, we know $P$ is perimeter minimizer in $\mathbb{R}^{n+1}$. We consider the blow-down limit of $P$ defined as follows. Define $P_r:=\frac{1}{r}P$. Up to a subsequence, $P_r$ converges to a set $P_\infty$ in $L^1_{\mathrm{loc}}(\mathbb{R}^{n+1})$. By compactness, $P_\infty$ is a perimeter minimizer in $\mathbb{R}^{n+1}$. The boundary of $P_r$, denoted by $M_r$, converges to the boundary of $P_\infty$ in the varifold sense; denote the limit by $V_\infty$.

By the previous regularity result, we know $\mathrm{sing}\,\|V_\infty\|$ is empty or discrete. If it is empty, then $V_\infty$ is a density-one hyperplane in $\mathbb{R}^{n+1}$. If it is discrete, then $V_\infty$ can only have an isolated singular point at the vertex $0$, since it is a stationary cone.

Now, we consider $P'_r:=P_r+e_{n+1}$. This is again a perimeter minimizer in $\mathbb{R}^{n+1}$, and we have $P_r \subset P'_r$. Up to a subsequence, $P'_r$ converges to a set $P'_\infty$, and $P'_\infty = P_\infty + e_{n+1}$. In particular, $P_\infty \subset P'_\infty$. By the strong maximum principle, we have either $P_\infty = P'_\infty$, or their boundaries are disjoint. In the first case, we are done, since $V_\infty$ is translation invariant along $e_{n+1}$ direction, it can only be a density-one hyperplane in $\mathbb{R}^{n+1}$. In the second case, we know that $V_\infty+\lambda e_{n+1}$ is disjoint from $V_\infty$ for any $\lambda\gt{}0$. Hence, we can construct a positive Jacobi field $u$ on $M_\infty$ by $u:=\left< \nu_{M_\infty}, e_{n+1} \right>$ where $M_\infty:=\mathrm{reg}\|V_\infty\|$. Now, for the function $V(r):=\int_{ \Sigma} \varphi(x) u(r x) d\Sigma(x)$, we have

\[ V(r)\leq \int_{ \Sigma} \varphi(x) \leq C \]

where $C$ is independent of $r$. On the other hand, we have $\kappa_7=2$, and hence

\[ V(r)\geq \frac{V(1)}{r^2} \text{ for } 0\lt{}r\lt{}1, \]

which implies $V(r)\to +\infty$ as $r\to 0$. This contradiction implies $\Sigma$ is totally geodesic, and hence $M_\infty$ is a density-one hyperplane in $\mathbb{R}^{n+1}$. ◻

Proof. By Allard’s regularity theorem, $M_r$ converges smoothly to a minimal hypersurface $M_\infty$ in $\mathbb{R}^{n+1}$.

In particular, if we denote $A_r$ to be the second fundamental form of $M_r$, then for any fixed $x\in M=M_1$, we have

\[ A_r(\frac{x}{r})\to A_\infty(0)=0 \]

On the other hand, we have

\[ A_r(\frac{x}{r})=rA(x) \]

which implies $A(x)=0$. Hence, $M$ is flat, so $u$ is an affine function. ◻

Properties of Minimal Submanifolds in R^N

Mon, 29 Jun 2026 00:00:00 +0000

Proposition 1.4.1

Let $\Sigma^n$ be a submanifold in $\mathbb{R}^N$ with position vector $x = (x_1, \ldots, x_N)$. Then its mean curvature vector $\vec{H}$ is given by

\[ \Delta^\Sigma x = \vec{H}. \]

Proof. Let $\{e_1, \ldots, e_n\}$ be an orthonormal basis for $T\Sigma$. The Laplacian of the coordinate $x_k$ on $\Sigma$ is given by

\[ \Delta^\Sigma x_k = \sum_{i=1}^n D_{e_i} D_{e_i} x_k = \sum_{i=1}^{n} D_{e_i} (e_i \cdot \frac{\partial }{\partial x_k}) = (D_{e_i}e_i)\cdot \frac{\partial }{\partial x_k} = \vec{H} \cdot \frac{\partial }{\partial x_k}, \]

Hence

\[ \Delta^\Sigma x = \vec{H}. \]

◻

Proposition 1.4.2

Let $\Sigma^n$ be a submanifold in $\mathbb{R}^N$ with position vector $x = (x_1, \ldots, x_N)$. Then $\Sigma$ is minimal if and only if each coordinate function $x_i$ is harmonic on $\Sigma$, i.e., $\Delta^\Sigma x_i = 0$ for all $i = 1, \ldots, N$.

Proposition 1.4.3

Suppose $\Sigma^n$ is a minimal submanifold in $\mathbb{R}^{n+1}$. Then $\nu\cdot \frac{\partial }{\partial x_i}$ satisfies the following Jacobi equation:

\[ \Delta^\Sigma (\nu\cdot \frac{\partial }{\partial x_i}) + |A|^{2}(\nu\cdot \frac{\partial }{\partial x_i}) = 0. \]

We may write this more concisely as

\[ L\nu=0. \]

Proof. This is because the flow generated by $\frac{\partial }{\partial x_i}$ is a family of isometries of $\mathbb{R}^{n+1}$, so $\frac{d}{dt}H=0$ for the corresponding variation. By the variation formula for the mean curvature, we have

\[ 0=\frac{d}{dt}H = -\Delta^\Sigma (\nu\cdot \frac{\partial }{\partial x_i}) - |A|^{2}(\nu\cdot \frac{\partial }{\partial x_i}). \]

◻

The Positive Mass Theorem and the Reduction to Geroch’s Conjecture

Mon, 29 Jun 2026 00:00:00 +0000

We now explain how the Riemannian positive mass theorem is reduced to Corollary 6.3.2. We state the time-symmetric version, which is the one governed by scalar curvature.

The time-symmetric, or Riemannian, positive mass theorem was first proved by Schoen–Yau by the minimal-hypersurface method [SY79b, SY81]. In the smooth dimension-reduction form of their argument, the regularity theory for area-minimizing hypersurfaces gives the theorem for $3\leq n\leq7$. Witten then gave a spinorial proof for spin asymptotically flat manifolds, without this regularity dimension restriction [Wit81].

The later history is largely about removing the non-spin dimension restriction. Schoen–Yau proposed an all-dimensional singular minimal-slicing approach [SY19], while Lohkamp’s cut-off/compactification observation relates the Riemannian PMT to Geroch torus rigidity; this is the reduction used below. The generic-regularity work culminating in Chodosh–Mantoulidis–Schulze–Wang pushes the minimal-hypersurface/Lohkamp route to dimension $11$ [CMSW25]. Bi–Hao–He–Shi–Zhu then proved the Riemannian PMT up to dimension $19$ [BHH+26], and Brendle–Wang subsequently gave a dimension descent scheme which closes the Riemannian theorem in arbitrary dimension [BW26a]. They also derived the spacetime positive energy theorem in arbitrary dimension from the Riemannian theorem and Jang-type arguments [BW26b].

There are also alternative proofs and stability directions which are useful to keep mentally separate from the reduction below. In dimension three, Bray–Kazaras–Khuri–Stern gave a harmonic-function proof of the Riemannian PMT [BKKS22]. Stability versions ask whether small ADM mass forces the geometry to be close to Euclidean space. Lee–Sormani proved pointed intrinsic-flat stability for rotationally symmetric asymptotically flat manifolds [LS14]. Huang–Lee–Sormani proved pointed intrinsic-flat stability for graphical hypersurfaces in Euclidean space under natural technical hypotheses [HLS17]. Dong–Song proved a three-dimensional stability theorem in the general asymptotically flat setting: if the mass of a chosen end tends to zero, then after removing subsets whose boundary areas tend to zero, the remaining spaces converge to Euclidean 3-space in the pointed measured Gromov–Hausdorff topology [DS25]. The proof below records the classical smooth Schoen–Yau/Lohkamp/Geroch mechanism, because that is the version whose topology is most visible from stable minimal hypersurfaces.

Theorem 6.4.1

Let $3\leq n\leq7$. Let $(M^n,g)$ be a complete one-ended asymptotically flat Riemannian manifold with nonnegative scalar curvature. Then the ADM mass is nonnegative. If the mass is zero and the usual rigidity hypotheses hold, then $(M,g)$ is isometric to Euclidean space.

For a manifold with several asymptotically flat ends, the same conclusion is applied to one end at a time after the standard reduction which compactifies or fills the other ends without changing the sign of the chosen mass. We focus on the one-ended case because it contains the topological argument.

Recall the ADM mass of an end with asymptotically flat coordinates $x$:

\[ m_{\operatorname{ADM}}(g) = \frac{1}{2(n-1)\omega_{n-1}} \lim_{r\to\infty} \int_{S_r} (\partial_jg_{ij}-\partial_ig_{jj})\nu^i\,d\sigma . \]

The nonnegativity part is the one whose topology is most transparent.

Definition 6.4.2

Fix an asymptotically flat end and write its coordinate chart as

\[ x=(x^1,\ldots,x^n):M\setminus K\to \mathbb{R}^n\setminus B_R, \qquad r=|x|. \]

The symbol $\delta$ denotes the Euclidean metric in these coordinates. For a tensor $T$ on the end and a number $\tau\gt{}0$, we use

\[ \|T\|_{L^p_{-\tau}}^p := \int_{M\setminus K} \bigl|(1+r)^\tau T\bigr|^p(1+r)^{-n}\,dx, \]

and

\[ \|T\|_{W^{k,p}_{-\tau}} := \sum_{j=0}^k \|\partial^jT\|_{L^p_{-\tau-j}}. \]

Thus $T\in W^{k,p}_{-\tau}$ means that $T$ decays like $r^{-\tau}$ and its $j$-th coordinate derivatives decay like $r^{-\tau-j}$ in weighted $L^p$ sense.

In particular, an AF metric of Sobolev type $(p,q)$ means

\[ g_{ij}-\delta_{ij}\in W^{2,p}_{-q}, \qquad p\gt{}n, \qquad q\gt{}\frac{n-2}{2}, \]

together with the integrability condition $R_g\in L^1$ when the ADM mass is used. The condition $p\gt{}n$ gives enough Sobolev embedding to read the metric and first derivatives pointwise, while $q\gt{}\frac{n-2}{2}$ is the decay threshold which makes the ADM mass stable under the density deformation.

If one starts instead from the classical pointwise AF assumption of order $\tau$,

\[ \partial^\alpha(g_{ij}-\delta_{ij}) = O(r^{-\tau-|\alpha|}), \qquad |\alpha|\leq2, \]

then the weighted Sobolev hypothesis follows with every smaller decay rate $q\lt{}\tau$. Indeed, for $|\alpha|\leq2$,

\[ \begin{aligned} \|\partial^\alpha(g-\delta)\|_{L^p_{-q-|\alpha|}}^p &\leq C\int_R^\infty r^{p(q+|\alpha|)} r^{-p(\tau+|\alpha|)} \frac{dr}{r} \\ &= C\int_R^\infty r^{-p(\tau-q)-1}\,dr \lt{}\infty . \end{aligned} \]

Thus a $C^2$ AF end of order $\tau\gt{}\frac{n-2}{2}$ gives the Sobolev assumption needed below after choosing $q$ with $\frac{n-2}{2}\lt{}q\lt{}\tau$. Many analytic statements of the density theorem take this Sobolev condition as the definition of asymptotic flatness.

There are two different ideas which should not be conflated. The original Schoen–Yau proof of the positive mass theorem uses a noncompact minimal hypersurface produced by a limiting Plateau argument. The more topological shortcut described below—cutting off a negative-mass end, compactifying it to a torus, and then contradicting torus rigidity—is usually attributed to Lohkamp’s compactification observation; see [Loh99]. Modern accounts phrase this as follows: using Lohkamp’s idea, one can reduce the Riemannian positive mass theorem to the impossibility of $R\gt{}0$ on $\mathbb{T}^n\#X^n$; see the discussion in [LUY24].

Theorem 6.4.3

Let $(M^n,g)$ be complete, one-ended, and asymptotically flat, with $R_g\geq0$ and negative ADM mass $m\lt{}0$. Assume the usual decay for which the ADM mass and the weighted elliptic theory are valid, including $R_g\in L^1$; for instance one may work with the Sobolev AF condition in Definition 6.4.2. Then, for every sufficiently small $\varepsilon\gt{}0$, there is a complete asymptotically flat metric $g_\varepsilon$ with the following properties:

\[ R_{g_\varepsilon}\geq0,\qquad |m_{\operatorname{ADM}}(g_\varepsilon)-m|\lt{}\varepsilon, \]

and on some exterior coordinate region $\{r\geq R_\varepsilon\}$,

\[ g_\varepsilon = u_\varepsilon^{\frac{4}{n-2}}\delta, \qquad \Delta^\delta u_\varepsilon=0, \qquad u_\varepsilon = 1+a_\varepsilon r^{2-n}+O_\infty(r^{1-n}). \]

Moreover

\[ m_{\operatorname{ADM}}(g_\varepsilon)=2a_\varepsilon. \]

Thus, if $\varepsilon\lt{}-m/2$, then $a_\varepsilon\lt{}0$. In the positive-mass contradiction argument, we may therefore replace the original metric by one which is conformally flat and scalar-flat near infinity, with negative harmonic mass coefficient.

Here the notation in the conclusion is as follows. The function $u_\varepsilon$ is a positive harmonic function on the exterior Euclidean coordinate region, and $a_\varepsilon$ is the coefficient of its leading $r^{2-n}$ term. The notation $O_\infty(r^\beta)$ is a shorthand for “big-$O$ with all derivatives”. More precisely, if $E=O_\infty(r^\beta)$, then for every multi-index $\alpha$ there is a constant $C_\alpha$ such that, in the chosen exterior coordinates,

\[ |\partial^\alpha E| \leq C_\alpha r^{\beta-|\alpha|} \qquad\text{for }r\text{ large}. \]

Thus $O_\infty(r^{1-n})$ means not only that the error itself is $O(r^{1-n})$, but also that every coordinate derivative has the corresponding differentiated decay; for example

\[ \partial^\alpha O_\infty(r^{1-n}) = O(r^{1-n-|\alpha|}). \]

This is stronger than ordinary $O(r^{1-n})$ notation and is the form needed when differentiating the expansion in the ADM mass or scalar-curvature computations. The equality $m_{\operatorname{ADM}}(g_\varepsilon)=2a_\varepsilon$ is the standard ADM normalization for a conformally flat end $u_\varepsilon^{4/(n-2)}\delta$.

Proof. This is the standard density step in the positive mass theorem; see [Bar86, LLU23]. We spell out the mechanism because it is the analytic part which precedes Lohkamp’s compactification.

Choose a large radial cut-off $\chi_\lambda$ on the asymptotically flat end, with $\chi_\lambda=1$ for $r\leq\lambda$ and $\chi_\lambda=0$ for $r\geq2\lambda$. Let $g_{\mathrm{Euc}}=\delta$ be the Euclidean metric in the AF coordinates and set

\[ g_\lambda = \chi_\lambda g+(1-\chi_\lambda)g_{\mathrm{Euc}} \]

on the end, extending it by $g$ on the compact part. Thus $g_\lambda=g$ on a large compact set and $g_\lambda=\delta$ for $r\geq2\lambda$. The only scalar curvature error is in the annulus $\{\lambda\leq r\leq2\lambda\}$. Define

\[ V_\lambda:=R_{g_\lambda}-\chi_\lambda R_g . \]

The basic estimate is

\begin{equation} \tag{6.4.1} \label{eq:density-error-small} \|V_\lambda\|_{L^p_{-q'-2}} + \|V_\lambda\|_{L^{n/2}} + \|V_\lambda\|_{L^{2n/(n+2)}} \to0 \end{equation}

for every $q'\lt{}q$ with $\frac{n-2}{2}\lt{}q'\lt{}n-2$. The point is that the linear part of the scalar curvature at the Euclidean metric is

\[ DR_\delta(k)=\partial_i\partial_jk_{ij}-\partial_j\partial_jk_{ii}. \]

When $k_\lambda=\chi_\lambda(g-\delta)$, the terms with two derivatives on $g-\delta$ are exactly $\chi_\lambda DR_\delta(g-\delta)$; the remaining terms contain at least one derivative of $\chi_\lambda$ and are supported in the annulus. Since $|\nabla^k\chi_\lambda|\leq C\lambda^{-k}$ there, and $g-\delta\in W^{2,p}_{-q}$, these commutator terms go to zero in the weighted norms above. The nonlinear terms in $R_{g_\lambda}$ contain either $(g_\lambda-\delta)\partial^2g_\lambda$ or $(\partial g_\lambda)^2$, and have the same decay. This proves (6.4.1).

For $\lambda$ large, solve the conformal correction equation

\begin{equation} \tag{6.4.2} \label{eq:density-conformal-correction} -c_n\Delta^{g_\lambda} w_\lambda+V_\lambda w_\lambda=0, \qquad w_\lambda\to1 \quad\text{on the chosen end}, \end{equation}

where $c_n=\frac{4(n-1)}{n-2}$. The weighted Fredholm estimate for the Laplacian on an asymptotically flat end, together with the smallness of $V_\lambda$ in (6.4.1), gives a unique solution with

\[ w_\lambda-1\in W^{2,p}_{-q'},\qquad \|w_\lambda-1\|_{W^{2,p}_{-q'}}\to0 . \]

Equivalently, writing $w_\lambda=1+\eta_\lambda$, the equation is

\[ \bigl(-c_n\Delta^{g_\lambda}+V_\lambda\bigr)\eta_\lambda = -V_\lambda. \]

The operator on the left is a small perturbation of the AF Laplacian $-c_n\Delta^{g_\lambda}:W^{2,p}_{-q'}\to L^p_{-q'-2}$, which is an isomorphism for $0\lt{}q'\lt{}n-2$. Thus a Neumann-series/Fredholm argument solves for $\eta_\lambda$ and gives the norm estimate above. This is the only analytic input in the deformation step. Since $p\gt{}n$, weighted Sobolev embedding gives $\|w_\lambda-1\|_{C^0}\to0$; after increasing $\lambda$ we therefore have $\frac12\leq w_\lambda\leq2$. Define

\[ \tilde g_\lambda:=w_\lambda^{\frac{4}{n-2}}g_\lambda . \]

The conformal scalar curvature formula gives

\[ \begin{aligned} R_{\tilde g_\lambda} &= w_\lambda^{-\frac{n+2}{n-2}} \bigl(-c_n\Delta^{g_\lambda}w_\lambda+R_{g_\lambda}w_\lambda\bigr)\\ &= w_\lambda^{-\frac{n+2}{n-2}} (R_{g_\lambda}-V_\lambda)w_\lambda = \chi_\lambda R_g\,w_\lambda^{-\frac{4}{n-2}} \geq0. \end{aligned} \]

Thus the scalar curvature sign is preserved. Since $w_\lambda$ stays uniformly bounded above and below and $g_\lambda$ is uniformly equivalent to $g$, the new metric is complete.

On the region $r\geq2\lambda$, we have $g_\lambda=\delta$ and $V_\lambda=0$, so (6.4.2) becomes

\[ \Delta^\delta w_\lambda=0 . \]

Therefore $\tilde g_\lambda$ is harmonically flat there:

\[ \tilde g_\lambda=w_\lambda^{\frac{4}{n-2}}\delta, \qquad w_\lambda=1+A_\lambda r^{2-n}+O_\infty(r^{1-n}). \]

Here the “conformal mass formula” is the following elementary consequence of the ADM boundary integral. If an AF metric $g_0$ has mass $m_{\rm ADM}(g_0)$ and

\[ \hat g=u^{\frac{4}{n-2}}g_0, \qquad u=1+A r^{2-n}+O_\infty(r^{1-n}), \]

then

\begin{equation} \tag{6.4.3} \label{eq:conformal-mass-adds-two-A} m_{\operatorname{ADM}}(\hat g) = m_{\operatorname{ADM}}(g_0)+2A . \end{equation}

Indeed, in the flat coordinates of the end, $u^{4/(n-2)}=1+\frac{4A}{n-2}r^{2-n}+O_\infty(r^{1-n})$, so the extra contribution to the ADM integrand is

\[ \partial_j\bigl((u^{4/(n-2)}-1)\delta_{ij}\bigr) - \partial_i\bigl((u^{4/(n-2)}-1)\delta_{jj}\bigr) = -(n-1)\partial_i(u^{4/(n-2)}), \]

and the boundary integral gives $2A$.

Applying (6.4.3) with $g_0=g_\lambda$ gives

\[ m_{\operatorname{ADM}}(\tilde g_\lambda) - m_{\operatorname{ADM}}(g_\lambda) = 2A_\lambda . \]

The equation for $w_\lambda$ then identifies this coefficient. Integrating

\[ -c_n\Delta^{g_\lambda}w_\lambda+V_\lambda w_\lambda=0 \]

over a large coordinate ball and letting the radius tend to infinity, using that $g_\lambda=\delta$ near infinity, gives

\[ 0 = -c_n\lim_{r\to\infty}\int_{S_r}\partial_\nu w_\lambda\,d\sigma + \int_M V_\lambda w_\lambda\,d\mu_{g_\lambda}. \]

Since $\partial_\nu w_\lambda=-(n-2)A_\lambda r^{1-n}+O(r^{-n})$ and $c_n=\frac{4(n-1)}{n-2}$, this becomes

\[ 2A_\lambda = -\frac{1}{2(n-1)\omega_{n-1}} \int_M V_\lambda w_\lambda\,d\mu_{g_\lambda}. \]

Hence

\[ m_{\operatorname{ADM}}(\tilde g_\lambda) - m_{\operatorname{ADM}}(g_\lambda) = -\frac{1}{2(n-1)\omega_{n-1}} \int_M V_\lambda w_\lambda\,d\mu_{g_\lambda}, \]

Here $g_\lambda$ is exactly Euclidean at infinity, so $m_{\operatorname{ADM}}(g_\lambda)=0$; the integral above is precisely what produces the new harmonic mass coefficient.

To see that this new mass is close to the original mass, one uses the standard mass-continuity lemma in the density theorem. The construction gives $\tilde g_\lambda-g\to0$ in $W^{2,p}_{-q'}$ for every $\frac{n-2}{2}\lt{}q'\lt{}n-2$, and the scalar curvatures also converge in $L^1$:

\[ R_{\tilde g_\lambda}-R_g = \chi_\lambda R_g \bigl(w_\lambda^{-\frac{4}{n-2}}-1\bigr) + (\chi_\lambda-1)R_g . \]

The first term goes to zero because $w_\lambda\to1$ uniformly and $R_g$ is integrable on the AF end; the second goes to zero because its support escapes to infinity. The ADM boundary integral is continuous under this $W^{2,p}_{-q'}$ convergence together with $L^1$ scalar-curvature convergence: this is the usual Bartnik density mass lemma, obtained by writing the mass integrand as the Euclidean linearization of scalar curvature plus quadratic terms, whose tails are integrable when $q'\gt{}\frac{n-2}{2}$. Hence

\[ m_{\operatorname{ADM}}(\tilde g_\lambda)\to m_{\operatorname{ADM}}(g)=m. \]

Choosing $\lambda$ large, the mass of $\tilde g_\lambda$ therefore differs from $m$ by less than $\varepsilon$.

It remains only to identify the coefficient $A_\lambda$ with the ADM mass. Set $v=w_\lambda^{4/(n-2)}$, so $\tilde g_{\lambda,ij}=v\delta_{ij}$ near infinity. Since

\[ v = 1+\frac{4A_\lambda}{n-2}r^{2-n}+O(r^{1-n}), \qquad \partial_\nu v=-4A_\lambda r^{1-n}+o(r^{1-n}), \]

we compute

\[ \partial_j\tilde g_{\lambda,ij} - \partial_i\tilde g_{\lambda,jj} = -(n-1)\partial_i v. \]

Substituting in the ADM formula yields

\[ m_{\operatorname{ADM}}(\tilde g_\lambda) = \frac{1}{2(n-1)\omega_{n-1}} \lim_{r\to\infty} \int_{S_r}4A_\lambda(n-1)r^{1-n}\,d\sigma = 2A_\lambda . \]

Taking $g_\varepsilon=\tilde g_\lambda$ and $a_\varepsilon=A_\lambda$ proves the theorem. ◻

Lemma 6.4.4

Assume that on an exterior coordinate region $\{r\geq R_0\}$ the metric is

\[ \begin{gathered} g=u^{\frac{4}{n-2}}\delta,\qquad R_g\geq0,\qquad a\lt{}0,\\ u=1+a r^{2-n}+o(r^{2-n}),\qquad \nabla u=-(n-2)a r^{1-n}\partial_r+o(r^{1-n}). \end{gathered} \]

Then, after increasing $R_0$ if necessary, one can replace $u$ outside a large compact set by a smooth positive function $\bar u$ such that

\[ \bar u=u \quad\text{near the compact core},\qquad \bar u\equiv c\gt{}0 \quad\text{near infinity},\qquad \Delta^\delta\bar u\leq0, \]

with strict inequality somewhere. Therefore

\[ \bar g=\bar u^{\frac{4}{n-2}}\delta \]

has nonnegative scalar curvature on the end, has positive scalar curvature somewhere in the transition region, and is exactly flat near infinity. Since $\bar u=u$ on a full neighborhood of the inner matching sphere, this replacement glues smoothly to the original metric on the compact part.

Proof. The conformal scalar curvature formula on the flat background gives

\[ R_g = -c_nu^{-\frac{n+2}{n-2}}\Delta^\delta u, \qquad c_n=\frac{4(n-1)}{n-2}. \]

Since $R_g\geq0$, the conformal factor is superharmonic:

\[ \Delta^\delta u\leq0. \]

The negative coefficient $a\lt{}0$ says that $u$ approaches $1$ from below. More precisely, for $r$ sufficiently large,

\[ \partial_r u=-(n-2)a r^{1-n}+o(r^{1-n})\gt{}0 . \]

After enlarging the compact core, choose $\varepsilon\gt{}0$ and radii $R_1\lt{}R_2$ so large that

\[ u\lt{}1-3\varepsilon \quad\text{on a collar of }S_{R_1}, \qquad u\gt{}1-\varepsilon \quad\text{for }r\geq R_2, \]

and such that $\partial_r u\gt{}0$ throughout $R_1\leq r\leq R_2$. Thus the transition set

\[ 1-3\varepsilon\lt{}u\lt{}1-\varepsilon \]

is contained in a compact annulus in the end, and $|\nabla u|\neq0$ somewhere inside this transition set.

Choose a smooth function $\Psi:\mathbb{R}\to\mathbb{R}$ with the following properties:

\[ \Psi(t)=t\quad\text{for }t\leq1-3\varepsilon,\qquad \Psi(t)=c\quad\text{for }t\geq1-\varepsilon, \]

where $c\gt{}0$, and

\[ 0\leq\Psi'\leq1,\qquad \Psi''\leq0, \]

with $\Psi''\lt{}0$ somewhere in $(1-3\varepsilon,1-\varepsilon)$. Such a function is obtained by choosing a smooth nonincreasing function $\theta:[1-3\varepsilon,1-\varepsilon]\to[0,1]$ which equals $1$ near the left endpoint and $0$ near the right endpoint, and then setting

\[ \Psi(t) = 1-3\varepsilon+\int_{1-3\varepsilon}^t\theta(s)\,ds \]

on the transition interval, with the two constant/identity extensions above.

Now set

\[ \bar u:=\Psi(u) \]

on the end, and keep $\bar u=u$ on the compact core. This is smooth across the inner matching region because $\Psi(t)=t$ wherever $u\leq1-3\varepsilon$. It is positive because $u\gt{}0$ and $c\gt{}0$, and it is constant equal to $c$ near infinity because $\Psi$ is constant for $t\geq1-\varepsilon$.

By the chain rule,

\[ \Delta^\delta\bar u = \Psi''(u)|\nabla u|^2+\Psi'(u)\Delta^\delta u \leq0. \]

The inequality is strict somewhere in the transition annulus: there $|\nabla u|\neq0$ at some point where $\Psi''(u)\lt{}0$. This proves the desired superharmonic cut-off. Notice that no spherical symmetry is used here; the only inputs are conformal flatness, superharmonicity, and the negative mass asymptotic.

Finally use the conformal scalar curvature formula with flat background metric. For any positive function $\phi$,

\[ R_{\phi^{4/(n-2)}\delta} = -c_n\phi^{-\frac{n+2}{n-2}}\Delta^\delta\phi, \qquad c_n=\frac{4(n-1)}{n-2}. \]

Applying this to $\phi=\bar u$ gives

\[ R_{\bar g} = -c_n\bar u^{-\frac{n+2}{n-2}}\Delta^\delta\bar u \geq0, \]

and it is positive somewhere. Since $\bar u$ is constant near infinity, $\bar g$ is flat there. ◻

Proposition 6.4.5

If a one-ended asymptotically flat manifold $(M^n,g)$, $3\leq n\leq7$, has nonnegative scalar curvature and negative ADM mass, then some closed manifold of the form

\[ \mathbb{T}^n\#Y \]

admits a positive-scalar-curvature metric.

Proof. First apply Theorem 6.4.3, choosing the deformation so that the mass remains negative. Then apply Lemma 6.4.4 to the harmonically flat end. We obtain a new complete metric, still denoted by $g$, with the following properties: $R_g\geq0$, $R_g\gt{}0$ somewhere in a compact annulus in the chosen end, and on $\{r\geq R_2\}$ the metric is

\[ g=c^{\frac{4}{n-2}}\delta \]

for a constant $c\gt{}0$.

Choose a coordinate radius $R$ with $R_0\lt{}R\lt{}R_2$, and let $K_R:=M\setminus\{r\gt{}R\}$ be the compact manifold obtained by cutting the chosen end at the coordinate sphere $S_R=\partial B_R$. Then choose $L\gt{}R_2$ and let

\[ Q_L=[-L,L]^n\subset\mathbb{R}^n. \]

Since every point of $\partial Q_L$ has Euclidean radius at least $L\gt{}R_2$, a whole collar of $\partial Q_L$ lies in the exactly flat region. Remove from $M$ the part of the chosen end outside $Q_L$. Equivalently, the remaining compact manifold with corners is

\[ W=K_R\cup_{S_R}(Q_L\setminus B_R), \]

where $Q_L\setminus B_R$ is read in the asymptotic coordinate chart. The outer boundary of this fundamental domain is the boundary of the cube $Q_L$.

Now identify opposite faces of $\partial Q_L$ by the translations

\[ (x^1,\ldots,x^i=L,\ldots,x^n) \sim (x^1,\ldots,x^i=-L,\ldots,x^n), \qquad i=1,\ldots,n. \]

These translations are isometries for the constant flat metric $c^{4/(n-2)}\delta$. Hence the metric descends smoothly across the identified faces. There is no corner singularity: the cube is only a fundamental domain for the standard smooth quotient $Q_L/\!\sim\,=\mathbb{T}^n$, and the metric is the translation-invariant flat metric in a neighborhood of the boundary faces.

Let $X$ be the closed quotient. We next identify its topology. Form the closed manifold

\[ Y:=K_R\cup_{S_R}B_R. \]

On the other hand, after the opposite faces of $Q_L$ are identified, $Q_L$ becomes $\mathbb{T}^n$, and the image of the interior ball $B_R$ is an embedded ball in this torus. Therefore

\[ X = K_R\cup_{S_R}\bigl((Q_L\setminus B_R)/\!\sim\bigr) \simeq Y\#\mathbb{T}^n . \]

This is the promised torus connected sum.

The scalar curvature statement is local, so it survives the quotient. Thus the induced metric on $X$ has $R\geq0$ everywhere and $R\gt{}0$ somewhere. By Lemma 6.1.2, the closed manifold $X$ admits a metric with $R\gt{}0$ everywhere. Since $X\simeq\mathbb{T}^n\#Y$, this proves the proposition. ◻

Proof. Suppose, to the contrary, that the mass is negative. By Proposition 6.4.5, some $\mathbb{T}^n\#Y$ admits a positive-scalar-curvature metric. This contradicts Corollary 6.3.2. Hence the ADM mass is nonnegative. ◻

The equality case is less topological but fits the same philosophy. If an asymptotically flat manifold with $R_g\geq0$ has zero mass and is not Euclidean, one uses a conformal/deformation argument to produce a new asymptotically flat metric with $R\geq0$ and strictly negative mass. This contradicts the nonnegativity just proved. In the original Schoen–Yau proof this is combined with the strong maximum principle and the regularity theory for the minimizing hypersurfaces.

The proof has three moving parts.

Stability plus the Gauss equation gives

\[ \int_\Sigma 2|\nabla\phi|^2+R_\Sigma\phi^2 \geq \int_\Sigma (R_N+|A|^2)\phi^2. \]

This is the whole reason positive scalar curvature descends to a stable minimal hypersurface.

A nonzero-degree map to a torus supplies cohomology classes whose cup product remains nonzero after passing to a Poincare-dual minimizing hypersurface.
A negative-mass end can be flattened to make a closed positive-scalar-curvature metric on $\mathbb{T}^n\#Y$, contradicting Geroch.

5. The decisive small-parameter calculation

Thu, 27 Aug 2026 00:00:00 +0000

The next phase no longer searched over unrelated formulas. We fixed

\[ \tag{5.1}\label{eq:pure-power-candidate} P(x)=\prod_{j=1}^m L_j^{2/m}, \qquad L_j=1-\ip{x}{p_j}, \]

and asked whether this balanced power was always viable away from its zeros when $k$ was sufficiently small.

5.1 The small-$k$ first-order expansion

Let $S=1-\nu_1^{2}-\cdots-\nu_\ell^{2}$. At a fixed point with $S\gt{}0$, direct expansion of (3.3) and (3.6) gives

\[ \tag{5.2}\label{eq:small-k-B-expansion} W=1+kw, \qquad w=P-\frac{1}{2}\sum_{i=1}^{\ell}\nu_iP_i, \qquad B_{aa}=1+k\mu_a+O(k^2), \]

where

\[ \mu_a =-\frac{1}{2}\sum_{i,j=1}^{\ell} P_{ij}\ip{e_i}{\tau_a}\ip{e_j}{\tau_a} -\frac{ \left(\sum_i\nu_i\ip{e_i}{\tau_a}\right) \left(\sum_jP_j\ip{e_j}{\tau_a}\right) }{S} -\frac{ P\left(\sum_i\nu_i\ip{e_i}{\tau_a}\right)^2 }{S^2}. \]

These were precisely the expressions implemented in the first-order numerical code: the input was $P$, its first two derivatives, the point $x$, and the frame coefficients $\ip{e_i}{\tau_a}$.

Put

\[ p_0(t)=\prod_{a=1}^{\ell}(t-\mu_a). \]

Substitution of (5.2) into the threshold polynomial (3.7) shows that

\[ \tag{5.3}\label{eq:critical-root} s=1+k\rho_++O(k^2), \qquad \rho_+\text{ is the largest real root of }p_0'. \]

The ambient dimension $n$ disappears from this first-order root; only the active rank $\ell$ remains. Define the gap coefficient

\[ \varrho=w-\rho_+. \]

Then

\[ \tag{5.4}\label{eq:normalized-small-k-threshold} W-s=k\varrho+O(k^2), \qquad \frac{s}{W}=1-k\varrho+O(k^2). \]

Consequently, the negative first-order term occurs in the normalized ratio $s/W$, whereas the unnormalized threshold in (5.3) has the positive first-order expansion. The clean sufficient condition is

\[ \class{key-formula-math}{\varrho\gt{}0.} \]

5.2 The first-order criteria in ranks two and three

For $\ell=2$, equation (5.3) immediately gives

\[ \rho_+=\frac{\mu_1+\mu_2}{2}, \qquad \varrho=w-\frac{\mu_1+\mu_2}{2}. \]

Equivalently, the first-order test is simply

\[ 2W-(B_{11}+B_{22})=2k\varrho+O(k^2)\gt{}0. \]

For the balanced product (5.1), direct substitution and a Cauchy–Schwarz estimate give $\varrho\gt{}0$ away from the zeros. Thus the balanced exponent $2/m$ passes the rank-two first-order test.

For $\ell=3$, put

\[ \sigma_1=\mu_1+\mu_2+\mu_3, \qquad \sigma_2=\mu_1\mu_2+\mu_1\mu_3+\mu_2\mu_3. \]

The critical-root equation becomes

\[ \tag{5.5}\label{eq:l3-rho-equation} 3\rho^2-2\sigma_1\rho+\sigma_2=0, \qquad \rho_+=\frac{\sigma_1+\sqrt{\sigma_1^2-3\sigma_2}}{3}. \]

Accordingly, $\varrho\gt{}0$ is equivalent to placing $w$ to the right of the larger root. A convenient pair of scalar checks is

\[ \tag{5.6}\label{eq:l3-sigma-test} 3w-\sigma_1\gt{}0, \qquad 3w^2-2\sigma_1w+\sigma_2\gt{}0. \]

This was the form used in the rank-three tests.

5.3 Rank-three scans and analytic confirmation

The numerical work tested (5.6) for coordinate, symmetric, clustered, tilted, and random prescribed directions. Every entry in Table 5.1 concerns the first-order gap

\[ \varrho =\left.\partial_k(W-s)\right|_{k=0^+} =w-\rho_+. \]

For each listed configuration, we numerically evaluated the normalized boundary gap $\varrho/P$ and the first-order gap $\varrho$. The randomized rows record separate scans for $m=2,\ldots,5$.

Pole configuration	$m$	$2/m$	Boundary $\min(\varrho/P)$	Sampled $\min\varrho$
Coordinate triple	3	$2/3$	$0.5006385$	$0.0999129$ ✓
Tetrahedral four	4	$1/2$	$0.5000182$	$0.3344756$ ✓
Octahedral six	6	$1/3$	$0.5000031$	$0.3157721$ ✓
Random four	4	$1/2$	$0.5009658$	$0.0426013$ ✓
Random six	6	$1/3$	$0.5000017$	$0.0427318$ ✓
Random eight	8	$1/4$	$0.5010803$	$0.1085162$ ✓
Randomized two-pole (60 tries)	2	$1$	---	$2.41\times10^{-1}$ ✓
Randomized three-pole (60 tries)	3	$2/3$	---	$2.06\times10^{-1}$ ✓
Randomized four-pole (60 tries)	4	$1/2$	---	$2.28\times10^{-1}$ ✓
Randomized five-pole (60 tries)	5	$2/5$	---	$2.92\times10^{-1}$ ✓

Table 5.1. Rank-three first-order scans for the balanced power product.

All four randomized families passed the first-order scans. The values in the table were numerical evidence only, but together they supported the stability of the balanced $2/m$ product away from the zero set. The computation also showed why samples taken too close to a zero could be misleading at fixed $k$: the first-order expansion is not uniform there.

After these scans, a detailed analytic calculation reduced (5.6) to a covariance estimate. It showed that $w$ lies strictly to the right of the larger root in (5.5) for every set of distinct prescribed directions. Consequently, $\varrho$ has a positive minimum on every compact set separated from the zeros. Continuity and (5.4) then give

\[ W-s\gt{}0 \]

there whenever $k\gt{}0$ is chosen sufficiently small.

This completed the away-from-zero part for $\ell\leq3$, but it did not solve the original problem. Near a prescribed direction,

\[ P(x)\sim C_jt^{2/m}, \qquad t=1-\ip{x}{p_j}, \qquad C_j\gt{}0, \]

and for every fixed positive $k$ the uniform lower bound could still fail in a thin neighborhood of that zero. Decreasing $k$ shrank the bad region but never removed it. The remaining task was therefore sharply defined: keep the successful $2/m$ tail away from the zeros and replace its local vanishing model. This led to the linear calculation in the next section.

Monotonicity Formula

Mon, 29 Jun 2026 00:00:00 +0000

Proposition 1.5.1

Suppose $\Sigma^n$ is a minimal submanifold in $\mathbb{R}^N$ and $x_0\in \mathbb{R}^N$. Then we have the following monotonicity formula:

\[ \frac{|\Sigma \cap B_\rho(x_0)|}{\rho^n} - \frac{|\Sigma \cap B_\sigma(x_0)|}{\sigma^n} = \int_{\Sigma \cap (B_\rho(x_0)\setminus B_\sigma(x_0))} \frac{|(x-x_0)^\perp|^{2}}{|x-x_0|^{n+2}} \, d\Sigma \]

Proof. Assume $x_0=0$, and choose the following vector field:

\[ V = \begin{cases} x\left( \frac{1}{|x|^n}-\frac{1}{\rho^n} \right), & \sigma\leq |x|\leq \rho\\ x\left( \frac{1}{\sigma^n}-\frac{1}{\rho^n} \right), & |x|\lt{}\sigma\\ \end{cases} \]

\[ \begin{aligned} 0={}&\int_{ \Sigma} \mathrm{div}^\Sigma V \, d\Sigma = \int_{ B_\sigma} \frac{n}{\sigma^n} d\Sigma - \int_{ \Sigma} \frac{n}{\rho^n}d\Sigma+\int_{ B_\rho\backslash B_\sigma}\mathrm{div}^\Sigma \frac{x}{|x|^n} d\Sigma\\ ={}& \frac{n|\Sigma\cap B_\sigma|}{\sigma^n} - \frac{n|\Sigma\cap B_\rho|}{\rho^n} + \int_{ B_\rho\backslash B_\sigma} \frac{n}{|x|^n}- \frac{n|x^T|^{2}}{|x|^{n+2}} d\Sigma\\ ={}& \frac{n|\Sigma\cap B_\sigma|}{\sigma^n} - \frac{n|\Sigma\cap B_\rho|}{\rho^n} + \int_{ B_\rho\backslash B_\sigma} \frac{n|x^\perp|^{2}}{|x|^{n+2}} d\Sigma. \end{aligned} \]

◻

Sketch of Wickramasekera’s Regularity Theorem

Mon, 29 Jun 2026 00:00:00 +0000

Theorem 4.5.1

Suppose $V_i$ is a sequence of stationary integral $n$-varifolds in $B_2^{n+1}(0)$ and $V_i$ also satisfies the following conditions:

$0\in \mathrm{spt}\|V_i\|$.
$\|V_i\|(B_2^{n+1}(0))\leq \Lambda$ for some constant $\Lambda\gt{}0$ independent of $i$.
(Stability) Each $V_i$ is stable in $B_2^{n+1}(0)$ on its regular set, i.e., for any $\phi \in C_c^1(\mathrm{reg}V_i)$,

\[ \int_{\mathrm{reg}V_i} |A_i|^2 \phi^2 d\|V_i\| \leq \int_{\mathrm{reg}V_i} |\nabla \phi|^2 d\|V_i\|. \]

(Alpha-Structural Hypothesis) There exists $\alpha\in (0,1)$ such that for each $i$, no point of $\mathrm{spt}\|V_i\|\cap B_1^{n+1}(0)$ has a neighborhood in which $\mathrm{spt}\|V_i\|$ is the union of three or more embedded $C^{1,\alpha}$ hypersurfaces-with-boundary meeting only along their common boundary.

We write the class of such varifolds as $\mathcal{V}_\alpha(\Lambda)$.

Theorem 4.5.2

Let $V_i\in \mathcal{V}_\alpha(\Lambda)$ such that $V_i \to q|B_2^n(0) \times \{ 0 \}$ in the varifold sense. Then, for $i$ large enough, we have

\[ V_{i}\left.\right|_{B_1^n \times \mathbb{R}} = \sum_{k=1}^{q} |\text{graph}u_{i,k}|, \]

with $u_{i,1}\leq \cdots \leq u_{i,q}$ and $u_{i,k}$ are smooth functions such that

\[ \|u_{i,k}\|_{C^{1,\alpha}(B_1^n)}^{2} \leq C \int_{B_2 } x_{n+1}^{2} d\|V_{i}\| \]

for some constant $C=C(n,\alpha,\Lambda)\gt{}0$ independent of $i$.

Theorem 4.5.3

Suppose $\boldsymbol{C}$ is a classical cone. There is no sequence of varifolds $V_i\in \mathcal{V}_\alpha(\Lambda)$ such that $V_i \to \boldsymbol{C}|_{B_2(0)}$ in the varifold sense.

Recall that a classical cone

\[ \boldsymbol{C}:= \sum_{i=1}^{N} q_i |H_i| \]

where $H_i$ are $n$-dimensional half-hyperplanes in $\mathbb{R}^{n+1}$ such that they contains origin and share the same boundary and $q_i$ are positive integers.

Sketch of the proof

The two theorems are proved simultaneously by induction on the density of planes and cones. Assume that the sheeting theorem holds for $q\leq q_0$ and that the minimal distance theorem holds for $\Theta(\boldsymbol{C},0)\leq q_0$.
Show that minimal distance theorem holds for $\Theta(\boldsymbol{C},0)\leq q_0+1$. It is enough to treat the densities $\Theta(\boldsymbol{C},0)=q_0+\frac{1}{2}$ and $q_0+1$.
Show that the sheeting theorem holds for $q=q_0+1$.
Dimension reduction argument plus the classification of stable cones implies that the singular set of $V_\infty$ has Hausdorff dimension at most $n-7$.

Proof of the minimal distance theorem

Assume the sheeting theorem holds for $q\leq q_0$ and the minimal distance theorem holds for $\Theta(\boldsymbol{C},0)\leq q_0$. Together with dimension reduction, this implies the desired regularity whenever $\Theta(\|V\|,X)\leq q_0$ for all $X\in B_2(0)$.
Suppose $V$ is sufficiently close to a classical cone $\boldsymbol{C}$ in the varifold sense and $\Theta(\|V\|,0)\geq \Theta(\|\boldsymbol{C}\|,0)$. Then, outside neighborhood of the singular set of $\boldsymbol{C}$, using the sheeting theorem, we can write $V$ as a union of smooth graphs over the half-hyperplanes in $\boldsymbol{C}$.
Establish the $L^{2}$-estimate for these graphs and key $L^{2}$ improvement in a smaller scale, following Simon’s cylindrical-singularity argument [Sim93].
By iteration, we can obtain the set $\{ \Theta(\|V\|,X)\geq \Theta(\|\boldsymbol{C}\|,0)\}$ is $C^{1,\alpha}$-regular for some $\alpha\gt{}0$, which implies $V$ has classical singularity at $0$, which is a contradiction.

Proof of the sheeting theorem

Same induction assumption as above.
Write $V_i$ as a union of Lipschitz graphs $u_{i,k}$ over the plane $\{ x_{n+1}=0 \}$ outside a small bad set.
Consider the blow-up class $v$, which is the limit of $E(V_i)^{-1}u_{i,k}$ where $E(V_i)$ is the $L^{2}$-excess of $V_i$.
($\star$) Prove that $v=(v_k)$ is harmonic.

First, we need a Lipschitz approximation for stationary varifolds.

Theorem 4.5.4

Fix $q\in\mathbb{Z}_{\geq 1}$ and $\sigma\in(0,1)$. Then there exists $\varepsilon_0=\varepsilon_0(n,q,\sigma)\in(0,1)$ such that the following holds. Let $V$ be a stationary integral $n$-varifold in $B_2^{n+1}(0)$ such that:

$\displaystyle \frac{1}{\omega_n 2^n}\,\|V\|\bigl(B_2^{n+1}(0)\bigr) \lt{} q+\frac12$ and $\displaystyle q-\frac12 \leq \frac{1}{\omega_n}\,\|V\|\bigl(\mathbb{R}\times B_1^n(0)\bigr) \lt{} q+\frac12$;
$\displaystyle \hat{E}^2(V) := \int_{\mathbb{R}\times B_1^n(0)} |x^1|^2\,\mathrm{d}\|V\|(X) \lt{} \varepsilon_0$.

Then there exists $\mathcal{H}^n$-measurable $\Sigma\subset B_\sigma^n(0)$ such that:

$\displaystyle \mathcal{H}^n(\Sigma)+\|V\|(\mathbb{R}\times\Sigma) \leq C\,\hat{E}_V^2$;
there exist Lipschitz functions $u_1,\ldots,u_q:B_\sigma^n(0)\to\mathbb{R}$ with $\operatorname{Lip}(u_j)\leq\frac12$, $u_1\leq\cdots\leq u_q$, $\displaystyle \sup_{B_\sigma^n(0)} |u_j| \leq C\,\hat{E}_V^{\frac{1}{n+1}}$, and

\[ V \mathbin{\llcorner} \bigl(\mathbb{R}\times (B_\sigma^n(0)\setminus\Sigma)\bigr) = \sum_{j=1}^q \bigl|\text{graph}(u_j|_{B_\sigma^n(0)\setminus\Sigma})\bigr|. \]

Here $C=C(n,q,\sigma)\in(0,\infty)$.

Moreover, we can establish the following estimate

\[ \int_{B_\sigma(0) } \sum_k |u_k|^{2}+|Du_k|^{2} \leq C E^{2}(V). \]

Definition 4.5.5

We consider $V_i$ as above such that $\hat{E}^{2}(V_i) \to 0^+$ and $\sigma_i \to 1$ as $i\to \infty$. The above estimate implies that $\hat{u}_k:=E^{-1}u_k$ satisfies

\[ \sum_{k}^{}\|u_k\|_{W^{1,2}(B_{\sigma_i}(0))}^{2} \leq C. \]

Up to a subsequence, we can find a weak limit function $v$. The blow-up class $\mathcal{B}_q$ is the collection of all such limits.

Theorem 4.5.6

The blow-up class $\mathcal{B}_q$ has the following properties:

$\mathcal{B}_q\subset W^{1,2}_{\mathrm{loc}}(B_1^n(0);\mathbb{R}^q)\cap L^2(B_1^n(0);\mathbb{R}^q)$.
If $v\in\mathcal{B}_q$, then $v^1\leq v^2\leq\cdots\leq v^q$.
If $v\in\mathcal{B}_q$, then $\Delta v_a=0$ in $B_1^n(0)$, where $v_a=q^{-1}\sum_{j=1}^q v^j$.
For each $v\in\mathcal{B}_q$ and each $z\in B_1^n(0)$, either ($\mathcal{B}$4 I**)** or ($\mathcal{B}$4 II**)** below holds:

($\mathcal{B}$4 I**)** (Hardt–Simon inequality [HS79]). For each $\rho\in\bigl(0,\tfrac{3}{8}(1-|z|)\bigr]$,

\[ \sum_{j=1}^q \int_{B_{\rho/2}^n(z)} R_z^{\,2-n}\left(\frac{\partial}{\partial R_z}\Bigl(\frac{v^j-v_a(z)}{R_z}\Bigr)\right)^{\!2}\,\mathrm{d}x \leq C\rho^{-n-2}\int_{B_\rho^n(z)}\bigl|v-\ell_{v,z}\bigr|^2\,\mathrm{d}x, \]

where $R_z(x)=|x-z|$, $\ell_{v,z}(x)=v_a(z)+Dv_a(z)\cdot(x-z)$, and $v-\ell_{v,z}=\bigl(v^1-\ell_{v,z},\ldots,v^q-\ell_{v,z}\bigr)$.

**(**<span class="math inline">&#92;&#40;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#66;&#125;&#92;&#41;</span><!-- -->4 *II***)** There exists <span class="math inline">&#92;&#40;&#92;&#115;&#105;&#103;&#109;&#97;&#61;&#92;&#115;&#105;&#103;&#109;&#97;&#40;&#122;&#41;&#92;&#105;&#110;&#40;&#48;&#44;&#49;&#45;&#124;&#122;&#124;&#93;&#92;&#41;</span> such that <span class="math inline">&#92;&#40;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#118;&#61;&#48;&#92;&#41;</span> in <span class="math inline">&#92;&#40;&#66;&#95;&#92;&#115;&#105;&#103;&#109;&#97;&#94;&#110;&#40;&#122;&#41;&#92;&#41;</span>.

If $v\in\mathcal{B}_q$, then:

($\mathcal{B}$5 I**)** $\displaystyle \tilde{v}_{z,\sigma}(\cdot)\equiv \|v(z+\sigma(\cdot))\|_{L^2(B_1^n(0))}^{-1}\,v(z+\sigma(\cdot))\in\mathcal{B}_q$ for each $z\in B_1^n(0)$ and $\sigma\in\bigl(0,\tfrac{3}{8}(1-|z|)\bigr]$ whenever $v\not\equiv 0$ in $B_\sigma^n(z)$;

($\mathcal{B}$5 II**)** $v\circ\gamma\in\mathcal{B}_q$ for each orthogonal rotation $\gamma$ of $\mathbb{R}^n$;

($\mathcal{B}$5 III**)** $\|v-\ell_v\|_{L^2(B_1^n(0))}^{-1}(v-\ell_v)\in\mathcal{B}_q$ whenever $v-\ell_v\not\equiv 0$ in $B_1^n(0)$, where $\ell_v(x)=v_a(0)+Dv_a(0)\cdot x$ for $x\in\mathbb{R}^n$ and $v-\ell_v=\bigl(v^1-\ell_v,\ldots,v^q-\ell_v\bigr)$.
If $\{v_k\}_{k=1}^\infty\subset\mathcal{B}_q$, then there exists a subsequence $\{k'\}$ of $\{k\}$ and a function $v\in\mathcal{B}_q$ such that $v_{k'}\to v$ locally in $L^2(B_1^n(0))$ and locally weakly in $W^{1,2}(B_1^n(0))$.
If $v\in\mathcal{B}_q$ is such that for each $j=1,\ldots,q$, there exist linear maps $L_1^j,L_2^j:\mathbb{R}^n\to\mathbb{R}$ with $v^j(x^2,y)=L_1^j(x^2,y)$ if $x^2\gt{}0$, $v^j(x^2,y)=L_2^j(x^2,y)$ if $x^2\leq 0$, and $L_1^j(0,y)=L_2^k(0,y)$ for all $1\leq j,k\leq q$ and $y\in\mathbb{R}^{n-1}$, where $(x^2,y)$ are coordinates on $\mathbb{R}^n$, then $v^1=v^2=\cdots=v^q=L$ for some linear map $L:\mathbb{R}^n\to\mathbb{R}$.

Proof.

This a consequence of the definition of the blow-up class.
This is based on $u_k\leq u_{k+1}$.
This is from the stationary property of $V_i$.
We can consider a new sequence $V_i'=(\eta_{z,\sigma})_{\#}V_i$, where its blow-up limit is $\tilde{v}_{z,\sigma}$. Similarly, we can consider $V_i'=\gamma_{\#}V_i$, where its blow-up limit is $v \circ \gamma$. Finally, the last property is from the rotation involving direction $e_{n+1}$.
This follows from the compactness of the blow-up sequence.
This is one of the key properties of the blow-up class, and the place that stable condition is used. The idea is, either we have good density points accumulating at a given point, which implies the Hardt–Simon inequality. Or we have a density gap, which by our induction assumption, implies that $v$ is harmonic.
This is the most technical part of the proof. It rules out the possibility that a blow-up class contains the singular model of a classical cone.

◻

Theorem 4.5.7

For any $v\in\mathcal{B}_q$, we have $v$ is smooth and harmonic.

Proof. This proof is similar to the classical dimensional reduction argument.

We define the “singular set” of $v$ as the set of points $x\in B_1^n(0)$ such that $\mathcal{B}4 \mathrm{I}$ holds.

First, we can study homogeneous degree-one functions in the blow-up class, namely tangent functions of a given function.

Claim. Each homogeneous 1 function in the blow-up class is a linear function.

Next, we can show that the “singular set” is empty by the dimensional reduction argument. ◻

After proving the blow-up class are all smooth and harmonic, we can show the sheeting theorem holds by using the standard regularity argument.

Proof of the property $\mathcal{B}7$

Suppose $v\in\mathcal{B}_q$ is such that for each $j=1,\ldots,q$, we can write

\[ v^j(x_1,y)=\lambda^j x_1 \text{ if } x_1\gt{}0, \text{ and } v^j(x_1,y)=\mu^j x_1 \text{ if } x_1\leq 0, \]

for some $\lambda^j,\mu^j\in\mathbb{R}$.

We define the cone $\boldsymbol{C}_k$ as the union of the graphs of the functions $\hat{E}_i \lambda^j x_1$ for $x_1\gt{}0$ and $\hat{E}_i \mu^j x_1$ for $x_1\leq 0$.

We can define the fine excess $E(V_i)$ as follows:

\[ E(V_i):= \int_{ \mathbb{R} \times B_1^n(0)} d^{2}(X,\boldsymbol{C}_i) d\|V_i\|(X). \]

Clearly, we have $E(V_i) \leq \hat{E}(V_i)$. But we expect that $E(V_i)$ is much smaller than $\hat{E}(V_i)$.

Indeed, we can establish the following under suitable assumptions.

\[ \lim \hat{E}^{-1}(V_i)E(V_i) = 0. \]

By the sheeting theorem, we can write $V_i$ as a union of smooth graphs over the cones $\boldsymbol{C}_i$ for $|x_1|\geq \sigma$.

Now, we define $h_i$ and $w_i$ the vector valued function such that

\[ V_i = \sum_{j=1}^{q} |h_i^j+\hat{E}_i \lambda^jx_1| + \sum_{j=1}^{q} |w_i^j+\hat{E}_i \mu^jx_1|, \]

and we expect that $h_i^j$ and $w_i^j$ much smaller than $\hat{E}_i$.

Establish the key $L^{2}$-estimate for $h_i^j,w_i^j$

Such estimate containst the following

\[ \int_{ B_{\frac{1}{2}}} \frac{|X^\bot|^{2}}{|X|^{n+2}}+\sum_{2\leq j\leq n}^{}|e_j^\bot|^{2}+\frac{d^{2}(X,\mathrm{spt}\,\|C_j\|)}{|X|^{n+2-\mu}} \leq C \hat{E}_i^2. \]

We can also establish the uniform bound for $E^{-1}(V_i)h_i^j$ and $E^{-1}(V_i)w_i^j$ and obtain the limit function $h$ and $w$. Such limit is called the fine blow-up limit.

Theorem 4.5.8

For the fine blow-up limit $h$ and $w$, we know $h,w$ are at least $C^{2}$ up to the boundary.

Proof. This relies on the key $L^{2}$ esimate using the fine excess.

First, we need to show the $C^{0,\alpha}$ estimate for $h,w$. We need to use the $L^{2}$-estimate to finish the proof.

Now, we need to show the $C^{2,\alpha}$ estimate for $h,w$. This is based on the stationary property of $V_i$ and the definition of the blow-ups, and the properties of harmonic functions.

Note that each $h,w$ are all harmonic in its interior, which is a consequence of the stationary property of $V_i$ and definition of the blow-ups.

The key is actually the smoothness up to the boundary. ◻

Improvement of the fine excess at a smaller scale

Using the properties of the fine blow-up limit, we can improve the fine excess at a smaller scale as follows.

Under suitable assumptions, with $\hat{E}(V_i)$ small enough, $\hat{E}^{-1}(V_i)E(V_i)$ is small enough, we can find a new cone $\boldsymbol{C}_i'$, such that if we apply a rotation, a scaling to the original varifold $V_i$, we can obtain a new varifold $V_i'$, such that the fine excess of $V_i'$ with respect to $\boldsymbol{C}_i'$ is much smaller than the fine excess of $V_i$ with respect to $\boldsymbol{C}_i$. One can think of this as follows. Under suitable assumptions, if one zooms in the original varifold $V_i$, it will look closer to a new cone $\boldsymbol{C}_i'$.

Iterative process

If such steps can be done infinitely many times for a given varifold $V_i$, we can obtain a limit cone $\boldsymbol{C}_{i,\infty}$. Note that the difference between $\boldsymbol{C}_{i,k}$ and $\boldsymbol{C}_{i,k+1}$ is much smaller than the difference between $\boldsymbol{C}_{i,k}$ and the plane $\{ x_{n+1}=0 \}$, because the fine excess is much smaller than the original excess. Hence $\boldsymbol{C}_{i,\infty}$ is still a classical cone. These steps also imply that we can find varifolds $V_{i,k}$ which converge to $\boldsymbol{C}_{i,\infty}$ in the varifold sense.

This is a contradiction to the minimal distance theorem.

The Horowitz–Myers Conjecture

Mon, 29 Jun 2026 00:00:00 +0000

The Horowitz–Myers conjecture [HM98] is a positive mass statement for spaces with negative cosmological constant. The point of this section is to explain the Riemannian version proved by Brendle–Hung [BH24], and then to outline how their systolic inequality proves it.

From AdS energy to a Riemannian inequality

Recall first the analogy with the positive mass theorem. For asymptotically flat initial data, Euclidean space is the reference geometry and the expected lower bound for the ADM mass is $0$. With a negative cosmological constant, the natural reference geometry is anti-de Sitter space. In spacetime language the Einstein equation is

\[ \mathrm{Ric}_{\mathbf g}-\frac12 R_{\mathbf g}\mathbf g+\Lambda\mathbf g =8\pi T. \]

For AdS geometry one takes $\Lambda\lt{}0$; in our normalization,

\[ \Lambda=-\frac{n(n-1)}2. \]

On a time-symmetric spacelike slice, the second fundamental form vanishes and the Hamiltonian constraint becomes a scalar-curvature condition. In vacuum it is

\[ R_g=-n(n-1), \]

and under the dominant energy condition it becomes

\[ R_g\geq -n(n-1). \]

For asymptotically hyperbolic data with spherical conformal infinity, the ground state is pure AdS, whose time-symmetric slice is hyperbolic space. The Horowitz–Myers phenomenon begins when the conformal infinity is toroidal. In that case the product hyperbolic end is not the expected lowest-energy geometry. One circle direction at infinity can fill in smoothly in the interior, producing the AdS soliton on $\mathbb R^2\times T^{n-2}$. This metric has negative mass relative to the product hyperbolic end. The conjecture says that, among metrics with the same toroidal asymptotics and $R_g\geq -n(n-1)$, the AdS soliton has the least possible mass.

The Brendle–Hung mass inequality

Let $\gamma$ be a flat metric on $S^1\times T^{n-2}$. The model end is

\[ \bar g=r^{-2}dr^2+r^2\gamma \]

on $(r_0,\infty)\times S^1\times T^{n-2}$. An asymptotically Horowitz–Myers end has an expansion

\[ g=\bar g+r^{2-n}Q+o(r^{2-n}), \]

where $Q$ is a symmetric $2$-tensor on $S^1\times T^{n-2}$. The quantity

\[ \int_{S^1\times T^{n-2}} n\operatorname{tr}_\gamma Q\,dV_\gamma \]

is the mass term in the normalization used here.

There is also a systolic quantity built into the toroidal end. Let $\xi:S^1\times T^{n-2}\to S^1$ be the circle projection and let $\Xi$ be the pullback of the volume form on $S^1$. Define

\[ \sigma = \inf\left\{ \operatorname{length}_\gamma(\alpha): \alpha\subset S^1\times T^{n-2}\text{ closed and } \int_\alpha \Xi\neq0 \right\}. \]

Thus $\sigma$ is the length of the shortest loop which winds nontrivially in the distinguished $S^1$-direction.

Theorem 6.5.1

Let $3\leq n\leq7$, and let $\gamma$ be a flat metric on $S^1\times T^{n-2}$. Let

\[ \bar{g}=r^{-2}dr^2+r^2\gamma \]

on $(r_0,\infty)\times S^1\times T^{n-2}$, and let $Q$ be a symmetric $2$-tensor on $S^1\times T^{n-2}$. Suppose $(N,g_N)$ is a smooth $n$-manifold such that:

$N\setminus E\cong (r_0,\infty)\times S^1\times T^{n-2}$ for some compact set $E\subset N$;
the $T^{n-2}$-projection on the end extends smoothly to a map $N\to T^{n-2}$;
on the end,

\[ |g_N-\bar g-r^{2-n}Q|_{\bar g}=o(r^{-n}), \qquad |\bar D(g_N-\bar g-r^{2-n}Q)|_{\bar g}=o(r^{-n}); \]

$R_{g_N}\geq -n(n-1)$.

Then

\[ \int_{S^1 \times T^{n-2}} n\,\operatorname{tr}_{\gamma}(Q)\,dV_\gamma \geq - \int_{S^1 \times T^{n-2}} \left( \frac{4\pi}{n\sigma} \right)^n dV_{\gamma}. \]

The left-hand side is the Horowitz–Myers mass term. The right-hand side is the mass of the corresponding AdS soliton. The rigidity statement, proved by Brendle–Hung in a subsequent work [BH25], says that equality forces the metric to be locally isometric to a Horowitz–Myers metric.

The systolic boundary inequality

The mass inequality is proved by cutting off the end and applying a sharp boundary inequality to the resulting compact manifold. We state that inequality in the compact form in which it is used.

Let $M$ be a compact, connected, orientable $n$-manifold with nonempty boundary. Suppose

\[ \xi:\partial M\to S^1,\qquad \theta=(\theta_1,\ldots,\theta_{n-2}):\partial M\to T^{n-2} \]

are smooth maps such that $(\xi,\theta):\partial M\to S^1\times T^{n-2}$ has nonzero degree. Let $\Xi=\xi^*(d\theta_{S^1})$, where $d\theta_{S^1}$ is the volume form of the circle. Similarly, let $\Theta_j=\theta_j^*(d\theta_j)$ denote the pulled-back volume forms from the circle factors of $T^{n-2}$. Define $\sigma$ to be the shortest length of a closed curve $\alpha\subset\partial M$ with $\int_\alpha\Xi\neq0$. Let $H_{\partial M}$ be the mean curvature of $\partial M$ with respect to the outward unit normal $\eta$.

Theorem 6.5.2

Let $3\leq n\leq7$, let $\beta\gt{}n$, and let $\varphi\in C^\infty(M)$. If

\[ -2 \Delta^M \varphi - \frac{\beta-n+1}{\beta-n} |\nabla^M \varphi|^2 + R_M + \beta(\beta-1) \geq 0, \]

then

\[ 2 \sigma^{\beta} \inf_{\partial M} \left( \langle \nabla^M \varphi, \eta \rangle + H_{\partial M} - (\beta-1) \right) \leq \left( \frac{4\pi}{\beta} \right)^\beta. \]

For the mass theorem, one only needs the scalar-curvature consequence obtained by taking $\varphi\equiv0$ and letting $\beta\to n$.

Corollary 6.5.3

With the same topological notation, if $R_M\geq -n(n-1)$, then

\[ 2 \sigma^{n} \inf_{\partial M} \left( H_{\partial M} - (n-1) \right) \leq \left( \frac{4\pi}{n} \right)^n. \]

The AdS soliton and sharpness

The sharp example is the Horowitz–Myers, or AdS soliton, metric. Geometrically one should picture the $S^1$-factor at infinity as a polar-angle direction which collapses smoothly in the interior. Thus the underlying manifold is $\mathbb R^2\times T^{n-2}$, while the conformal infinity is $S^1\times T^{n-2}$.

For simplicity, assume that $(S^1,g_{S^1})$ has length $4\pi/n$. On $(1,\infty)\times S^1\times T^{n-2}$, set

\[ g=\frac{1}{\rho^2(1-\rho^{-n})}d\rho^2 +\rho^2(1-\rho^{-n})g_{S^1}+\rho^2g_{T^{n-2}}. \]

With the substitution

\[ \rho=\left(\cosh \frac{n\tilde\rho}{2}\right)^{2/n}, \]

this becomes

\[ g=d\tilde\rho^2+ \left( \cosh \frac{n \tilde{\rho}}{2} \right)^{4/n} \left[ \left( \tanh \frac{n\tilde{\rho}}{2} \right)^2 g_{S^1} +g_{T^{n-2}} \right]. \]

Near $\tilde\rho=0$, the first two directions look like

\[ d\tilde\rho^2+\left(\frac{n\tilde\rho}{2}\right)^2g_{S^1}. \]

The choice $\operatorname{length}(S^1)=4\pi/n$ is exactly the no-cone-angle condition, so the metric extends smoothly across the collapsed circle. The resulting metric has scalar curvature

\[ R_g=-n(n-1). \]

To compare it with the asymptotic model, define $r$ by

\[ \rho^{n/2}=r^{n/2}\left(1+\frac14 r^{-n}\right). \]

Then

\[ \begin{aligned} g={}&r^{-2}dr^2 +r^2\left(1+\frac14r^{-n}\right)^{4/n-2} \left(1-\frac14r^{-n}\right)^2g_{S^1} +r^2\left(1+\frac14r^{-n}\right)^{4/n}g_{T^{n-2}}\\ ={}&r^{-2}dr^2 +r^2\left(1-\frac{n-1}{n}r^{-n}\right)g_{S^1} +r^2\left(1+\frac1n r^{-n}\right)g_{T^{n-2}} +o(r^{-n}). \end{aligned} \]

Thus the tensor $Q$ is

\[ Q=-\frac{n-1}{n}g_{S^1}+\frac{1}{n}g_{T^{n-2}}, \]

and

\[ \operatorname{tr}_\gamma Q=-\frac1n. \]

Since $\sigma=4\pi/n$, the right-hand side in Theorem 6.5.1 is exactly the mass of this metric. Hence the inequality is sharp.

How the Brendle–Hung proof works

We keep the proof in three parts. The first part converts the boundary inequality into the mass inequality. The second part proves the boundary inequality by dimension reduction. The final part is the two-dimensional endpoint, where the sharp constant is computed.

Step 1: from the boundary inequality to the mass inequality.

The goal is to prove Theorem 6.5.1:

\[ \int_{S^1\times T^{n-2}} n\operatorname{tr}_{\gamma}Q\,dV_{\gamma} \geq -\int_{S^1\times T^{n-2}} \left(\frac{4\pi}{n\sigma}\right)^n dV_{\gamma}. \]

The compact input is Corollary 6.5.3. Thus we need to cut off the end of $N$ in such a way that the boundary mean curvature sees the mass aspect.

Choose $u$ and a constant $\mu$ on $S^1\times T^{n-2}$ by

\[ \Delta^\gamma u+\frac n2\operatorname{tr}_{\gamma}Q+\mu=0, \qquad \int_{S^1\times T^{n-2}}u\,dV_{\gamma}=0. \]

Equivalently,

\[ \int_{S^1\times T^{n-2}} \left(n\operatorname{tr}_{\gamma}Q+2\mu\right)dV_{\gamma}=0. \]

For large $\widehat r$, cut off the end by the graph

\[ \widehat N = N\cap\{r\leq \widehat r+\widehat r^{3-n}\widehat u\}, \qquad \widehat u=u\circ\pi. \]

The reason for using this graph, rather than the coordinate torus $\{r=\widehat r\}$, is that the graph makes the first nontrivial term in the mean curvature constant. The two estimates one needs are

\[ D^2\widehat u = D_{\gamma}^2u -r^{-1}(dr\otimes d\widehat u+d\widehat u\otimes dr) +O(r^{-n-1}) \]

and

\[ D^2r=rg-\frac n2 r^{3-n}Q+o(r^{1-n}). \]

Substituting these into the level-set formula

\[ H_{\partial\widehat N} = \frac{\operatorname{tr}_{\partial\widehat N}D^2 (r-\widehat r^{3-n}\widehat u)} {|D(r-\widehat r^{3-n}\widehat u)|} \]

gives

\[ H_{\partial\widehat N} =(n-1)+\widehat r^{-n}\mu+o(\widehat r^{-n}). \]

Therefore Corollary 6.5.3 applied to $\widehat N$ gives

\[ 2\widehat\sigma^n\widehat r^{-n}\mu \leq \left(\frac{4\pi}{n}\right)^n+o(1), \]

where $\widehat\sigma$ is the boundary systole on $\partial\widehat N$. Since

\[ \widehat r^{-2}g_{\partial\widehat N}\to\gamma, \qquad \frac{\widehat\sigma}{\widehat r}\to\sigma, \]

we pass to the limit and recover the mass inequality. Equivalently, one may argue by contradiction: if the mass integral were too negative, then for some $\varepsilon\gt{}0$,

\[ 2(1-\varepsilon)^{n+1}\sigma^n\mu \geq \left(\frac{4\pi}{n}\right)^n, \]

whereas the boundary inequality gives

\[ 2(1-\varepsilon)\widehat\sigma^n\widehat r^{-n}\mu \leq \left(\frac{4\pi}{n}\right)^n \]

for large $\widehat r$. Hence $\widehat\sigma/\widehat r\leq (1-\varepsilon)\sigma$, contradicting $\widehat\sigma/\widehat r\to\sigma$.

Step 2: dimensional reduction for the systolic inequality.

The goal is now Theorem 6.5.2. The proof reduces the $n$-dimensional boundary inequality to a two-dimensional one by repeatedly taking free-boundary hypersurfaces which preserve the relevant cohomological information.

The inductive object is a pair $(\Sigma^k,\varphi_k)$, where $\Sigma^k$ has boundary and carries the nontrivial topological information coming from $\Xi,\Theta_1,\ldots,\Theta_{k-2}$. The pair is required to satisfy

\[ E_k(\varphi_k,g_k):= -2\Delta^{\Sigma^k}\varphi_k -\frac{\beta-k+1}{\beta-k} |\nabla^{\Sigma^k}\varphi_k|^2 +R_{\Sigma^k}+\beta(\beta-1)\geq0, \]

and the boundary term is

\[ B_k:=\partial_\eta\varphi_k+H_{\partial\Sigma^k}. \]

Starting with $\Sigma^n=M$ and $\varphi_n=\varphi$, the reduction step is:

Reduction proposition.

Assume $E_k(\varphi_k,g_k)\geq0$ and $B_k\geq0$. Then there is a compact free boundary hypersurface $\Sigma^{k-1}\subset\Sigma^k$, stable for the weighted area functional

\[ \mathcal A_k(S)=\int_S e^{\varphi_k}\,dA_{g_k}, \]

and a function $\varphi_{k-1}$ on $\Sigma^{k-1}$, such that

\[ E_{k-1}(\varphi_{k-1},g_{k-1})\geq0 \]

and

\[ \partial_\eta\varphi_{k-1} +H_{\partial\Sigma^{k-1}} = \partial_\eta\varphi_k+H_{\partial\Sigma^k}. \]

The hypersurface is chosen in the homology class detected by the remaining forms, so the relevant systole cannot decrease in the direction needed for the final inequality.

Let us indicate why the differential inequality is preserved. Write $\bar\Sigma=\Sigma^k$, $\widetilde\Sigma=\Sigma^{k-1}$, and $\bar\varphi=\varphi_k|_{\widetilde\Sigma}$. All geometric quantities below are computed in the original metric $g_k$. If $\widetilde\nu$ is the unit normal of $\widetilde\Sigma\subset\bar\Sigma$, then the second variation of $\mathcal A_k$ gives, for every smooth test function $\zeta$,

\[ \begin{aligned} 0\leq{}& \int_{\widetilde\Sigma} e^{\bar\varphi} \left( |\widetilde\nabla\zeta|^2 -\bigl(\overline{\operatorname{Ric}}(\widetilde\nu,\widetilde\nu) +|\widetilde A|^2\bigr)\zeta^2 +\bar{\nabla}^2\bar\varphi(\widetilde\nu,\widetilde\nu)\zeta^2 \right)\\ &\quad -\int_{\partial\widetilde\Sigma}e^{\bar\varphi} A_{\partial\bar\Sigma}(\widetilde\nu,\widetilde\nu)\zeta^2 . \end{aligned} \]

Here $\widetilde A$ is the second fundamental form of $\widetilde\Sigma\subset\bar\Sigma$. The usual Gauss equation and the free-boundary relation give

\[ |\widetilde A|^2+ \overline{\operatorname{Ric}}(\widetilde\nu,\widetilde\nu) = \frac12\left(\bar R-\widetilde R +|\widetilde A|^2+\widetilde H^2\right) \geq \frac12\left(\bar R-\widetilde R+\widetilde H^2\right), \]

and

\[ A_{\partial\bar\Sigma}(\widetilde\nu,\widetilde\nu) = H_{\partial\bar\Sigma}-H_{\partial\widetilde\Sigma}. \]

Thus the stability inequality may be weakened to

\[ \begin{aligned} 0\leq{}& \int_{\widetilde\Sigma} e^{\bar\varphi} \left( |\widetilde\nabla\zeta|^2 -\frac12(\bar R-\widetilde R+\widetilde H^2)\zeta^2 +\bar{\nabla}^2\bar\varphi(\widetilde\nu,\widetilde\nu)\zeta^2 \right)\\ &\quad -\int_{\partial\widetilde\Sigma}e^{\bar\varphi} (H_{\partial\bar\Sigma}-H_{\partial\widetilde\Sigma})\zeta^2 . \end{aligned} \]

By the first-eigenfunction argument for this Robin problem, there is a positive function $\omega$ on $\widetilde\Sigma$ and a number $\lambda\geq0$ such that

\[ \begin{aligned} &-\operatorname{div}^{\widetilde\Sigma} (e^{\bar\varphi}\widetilde\nabla\omega) -\frac12(\bar R-\widetilde R+\widetilde H^2) e^{\bar\varphi}\omega +e^{\bar\varphi} \bar{\nabla}^2\bar\varphi(\widetilde\nu,\widetilde\nu)\omega\\ &\qquad\qquad =\lambda e^{\bar\varphi}\omega\geq0, \end{aligned} \]

with boundary condition

\[ \partial_\eta\omega -(H_{\partial\bar\Sigma}-H_{\partial\widetilde\Sigma})\omega=0. \]

Define

\[ \widetilde\varphi=\varphi_{k-1}:=\bar\varphi+\log\omega. \]

The Schoen–Yau rearrangement of the stability inequality gives

\[ \begin{aligned} &-2\widetilde\Delta\widetilde\varphi+\widetilde R -|\widetilde\nabla\widetilde\varphi|^2 \\ &\qquad\geq -2\bar\Delta\bar\varphi+\bar R -|\bar\nabla\bar\varphi|^2 +|\widetilde\nabla\bar\varphi -\widetilde\nabla\widetilde\varphi|^2 . \end{aligned} \]

The only algebraic point is

\[ |a-b|^2+p|a|^2 \geq \frac{p}{1+p}|b|^2. \]

Taking $p=1/(\beta-k)$ gives

\[ |\widetilde\nabla\bar\varphi -\widetilde\nabla\widetilde\varphi|^2 +\frac{1}{\beta-k} |\widetilde\nabla\bar\varphi|^2 \geq \frac{1}{\beta-k+1} |\widetilde\nabla\widetilde\varphi|^2 . \]

Therefore

\[ \begin{aligned} &-2\widetilde\Delta\widetilde\varphi+\widetilde R -\left(1+\frac{1}{\beta-k+1}\right) |\widetilde\nabla\widetilde\varphi|^2+\beta(\beta-1)\\ &\qquad\geq -2\bar\Delta\bar\varphi+\bar R -\left(1+\frac{1}{\beta-k}\right) |\bar\nabla\bar\varphi|^2+\beta(\beta-1)\geq0. \end{aligned} \]

This is exactly the inequality $E_{k-1}\geq0$. The boundary equality follows directly from the Neumann condition for $\omega$:

\[ \partial_\eta\widetilde\varphi+H_{\partial\Sigma^{k-1}} = \partial_\eta\bar\varphi+H_{\partial\Sigma^k}. \]

Iterating the reduction either stops early, in which case the boundary infimum is already nonpositive and the desired inequality is immediate, or reaches a surface $(\Sigma^2,\varphi_2)$. In the latter case the boundary terms are the same along the construction and, if $\sigma_2$ is the corresponding systole on $\partial\Sigma^2$, then $\sigma\leq\sigma_2$. The two-dimensional estimate therefore gives

\[ \begin{aligned} &2\sigma^\beta \inf_{\partial M} \left(\partial_\eta\varphi+H_{\partial M}-(\beta-1)\right)\\ &\qquad\leq 2\sigma_2^\beta \inf_{\partial\Sigma^2} \left(\partial_\eta\varphi_2 +H_{\partial\Sigma^2}-(\beta-1)\right) \leq \left(\frac{4\pi}{\beta}\right)^\beta . \end{aligned} \]

Step 3: the two-dimensional endpoint and the monotonicity.

The dimension reduction leaves a sharp inequality on a surface. This is the only place where the numerical constant $(4\pi/\beta)^\beta$ is produced explicitly.

Theorem 6.5.4

Let $\Sigma$ be a compact connected orientable surface with nonempty boundary. Let $K$ be its Gaussian curvature, $\kappa$ the geodesic curvature of $\partial\Sigma$, and $\eta$ the outward unit normal. Suppose

\[ -2\Delta\psi-\frac{\beta-1}{\beta-2}|\nabla\psi|^2 +2K+\beta(\beta-1)\geq0. \]

Then:

If $\Sigma$ is diffeomorphic to a disk, then

\[ 2|\partial\Sigma|^\beta \inf_{\partial\Sigma} \left(\partial_\eta\psi+\kappa-(\beta-1)\right) \leq \left(\frac{4\pi}{\beta}\right)^\beta . \]

If $\Sigma$ is not diffeomorphic to a disk, then

\[ \inf_{\partial\Sigma} \left(\partial_\eta\psi+\kappa-(\beta-1)\right)\leq0. \]

The second case is proved by reducing further to a free-boundary geodesic. The main new estimate is the disk case.

For the disk case define the parallel domains

\[ \Omega_s=\{x\in\Sigma:d(x,\partial\Sigma)\gt{}s\}, \qquad \gamma_s=\partial\Omega_s, \qquad L(s)=\mathcal H^1(\gamma_s), \]

and let $l=\sup\{s:\Omega_s\neq\emptyset\}$. Put

\[ F(s)=\tanh\frac{\beta s}{2}, \qquad G(s)=\left(\cosh\frac{\beta s}{2}\right)^{\frac{2(\beta-1)}{\beta}}. \]

For almost every $s$, the curve $\gamma_s$ is piecewise smooth. If $\Gamma(s)$ denotes its total curvature, then the comparison geometry gives

\[ L'(s)\leq -\Gamma(s). \]

Since the connected components of $\Omega_s$ are disks, Gauss–Bonnet gives

\[ 2\pi\leq \Gamma(s)+\int_{\Omega_s}K. \]

Now define

\[ I(s)=2\pi-(\beta-1)F(l-s)L(s) +\int_{\Omega_s}(\Delta\psi-K). \]

The fundamental monotonicity statement is

\[ I'(s)-(\beta-1)F(l-s)I(s)\geq0 \qquad\text{for a.e. }s\in(0,l). \]

The computation is short enough to record the structure. Differentiate $I$, use $L'(s)\leq-\Gamma(s)$, use Gauss–Bonnet to replace $\Gamma(s)$, and use the scalar inequality to control the integral over $\gamma_s$. The last point is the elementary estimate

\[ \frac{\beta-1}{2(\beta-2)}|\nabla\psi|^2 +\frac{(\beta-1)(\beta-2)}{2}F^2(l-s) \geq (\beta-1)F(l-s)|\nabla\psi|. \]

These inequalities combine to give

\[ I'(s)\geq(\beta-1)F(l-s)I(s). \]

Since

\[ (\log G)'(s)=(\beta-1)F(s), \]

the equivalent formulation is

\[ J(s):=G(l-s)I(s) \qquad\Longrightarrow\qquad J'(s)\geq0. \]

This is the monotonicity one should remember.

Finally $J(0)\leq J(l)$. Using Gauss–Bonnet on the original disk, this gives

\[ 2\pi \geq G(l)\left( \int_{\partial\Sigma}(\partial_\eta\psi+\kappa) -(\beta-1)F(l)|\partial\Sigma| \right). \]

Let $\sigma=|\partial\Sigma|$ in the disk case. Write

\[ a:=\inf_{\partial\Sigma} \left(\partial_\eta\psi+\kappa-(\beta-1)\right). \]

The preceding inequality controls the boundary average, hence also the infimum:

\[ a \leq \frac{1}{\sigma}\int_{\partial\Sigma} (\partial_\eta\psi+\kappa-(\beta-1)) \leq \frac{2\pi}{\sigma G(l)}-(\beta-1)(1-F(l)). \]

Multiplying by $2\sigma^\beta$ gives

\[ 2\sigma^\beta a \leq \frac{4\pi\sigma^{\beta-1}}{G(l)} -2(\beta-1)(1-F(l))\sigma^\beta . \]

Since $0\leq F(l)\lt{}1$, we have

\[ 1-F(l)\geq \frac12(1-F^2(l)) =\frac12G(l)^{-\frac{\beta}{\beta-1}}. \]

Therefore

\[ 2\sigma^\beta a \leq \frac{4\pi\sigma^{\beta-1}}{G(l)} -(\beta-1)\frac{\sigma^\beta} {G(l)^{\frac{\beta}{\beta-1}}}. \]

Now set

\[ x=\frac{\sigma}{G(l)^{1/(\beta-1)}}. \]

The right-hand side becomes

\[ 4\pi x^{\beta-1}-(\beta-1)x^\beta . \]

This one-variable expression is maximized at $x=4\pi/\beta$, and its maximum is

\[ \left(\frac{4\pi}{\beta}\right)^\beta . \]

This proves Theorem 6.5.4. Step 2 gives Theorem 6.5.2, Corollary 6.5.3 follows by taking $\varphi\equiv0$ and $\beta\to n$, and Step 1 proves the Horowitz–Myers mass inequality.

6. The local repair and the final profile

Thu, 27 Aug 2026 00:00:00 +0000

6.1 The linear model near a zero

Once the balanced $2/m$-power had been shown to work away from its zeros, the remaining question was local. Near the prescribed direction $p_1$, put

\[ t=1-\ip{x}{p_1}, \qquad P(x)=t^\alpha P_{\mathrm{reg}}(x), \qquad P_{\mathrm{reg}}(p_1)\gt{}0. \]

The other pole factors are smooth and positive there, so the leading behavior is determined by the single exponent $\alpha$. This reduced the first question to a direct one-pole calculation: which powers retain a positive first-order gap as $t\downarrow0$?

The fixed-$k$ one-pole calculation ruled out exponents that were too small. If $C=P_{\mathrm{reg}}(p_1)\gt{}0$, its leading term is

\[ \tag{6.1}\label{eq:local-alpha-expansion} W-s =Ct^{\alpha-1} \left( \frac{\alpha\bigl((n-1)\alpha-(n-2)\bigr)}{2n} +O(t) \right). \]

Thus a strictly positive leading margin requires

\[ \alpha\gt{}\frac{n-2}{n-1}. \]

The natural dimension-independent choice is therefore $\alpha=1$, corresponding to the linear model

\[ \phi(t)\sim ct \qquad(t\downarrow0). \]

A direct substitution of this model restores the positive near-zero margin that was lost by the pure $t^{2/m}$ power.

6.2 The logarithmic transition and the rank threshold

The next question was whether the local exponent $1$ could be connected to the successful outer exponent $2/m$. This is a different $k\downarrow0$ calculation from the fixed-$k$ zero model above. An arbitrary smooth cutoff need not work, because its second derivative appears in the first-order threshold. For a positive profile $\phi$, we therefore introduced its logarithmic slope and curvature

\[ \xi=\frac{\dd\log\phi}{\dd\log t}, \qquad \eta=\frac{\dd\xi}{\dd\log t}. \]

Thus $\xi=1$ is the linear model, $\xi=a$ is a power $t^a$, and $\xi=0$ is a plateau. Applying the small-parameter calculation to the one-pole transition gives a second-order differential inequality for $(\xi,\eta)$. Here the ambient dimension $n$ disappears; only the active rank $\ell$ remains. The two endpoint regimes give the necessary conditions

\[ \eta\gt{}-\xi^2+\frac{3\ell-4}{2(\ell-1)}\xi-\frac{1}{2} \quad(\ell\geq2), \qquad \eta\gt{}\frac{\ell-3}{2(\ell-2)}\xi-\frac{1}{2}\xi^2 \qquad(\ell\geq3). \]

Setting $\eta=0$ shows that a terminal power $t^a$ must satisfy

\[ \tag{6.2}\label{eq:rank-threshold} a\gt{}\frac{\ell-3}{\ell-2} \qquad(\ell\geq3). \]

The rank dependence became transparent in this ODE.

Rank two. The radial condition, which is the full logarithmic condition in rank two, becomes

\[ \eta+\xi^2-\xi+\frac{1}{2}\gt{}0. \]

It allows $\xi$ to decrease from $1$ all the way to $0$ in a finite logarithmic interval. After rescaling, this transition can be placed in an arbitrarily small pole cap. Outside the caps every factor is constant, so $P$ itself is constant and the rank-two construction becomes especially simple.

Rank three. A convenient sufficient condition for the full one-pole ODE is

\[ 0\lt{}\xi\leq1, \qquad \eta\geq-c\xi^2, \qquad 0\lt{}c\lt{}\frac{7}{32}. \]

This condition prevents $\xi$ from reaching $0$ in finite logarithmic time, explaining the failure of the earlier linear-to-plateau attempts. It does, however, allow $\xi$ to decrease from $1$ to any fixed positive exponent, in particular to $2/m$. The logarithmic transition has a fixed finite length once $m$ is fixed; shrinking its scale places the entire transition in an arbitrarily small physical neighborhood of the prescribed direction.

Rank four. Condition (6.2) requires a terminal exponent strictly greater than $1/2$. Thus the balanced exponent $2/m$ has zero limiting margin when $m=4$, and it lies below the admissible range when $m\gt{}4$. This local obstruction already shows why the same test-function design encounters a genuine difficulty in a rank-four sheeting problem.

6.3 The final profile and its verification

For the rank-three construction, choose a small scale $T\gt{}0$ and a smooth nondecreasing profile such that

\[ \phi_T(t)= \begin{cases} T^{2/m-1}t, & 0\leq t\leq T,\\ C_m t^{2/m}, & t\geq e^L T, \end{cases}, \]

where $L\lt{}\infty$ is the logarithmic length of the chosen transition. It may be arranged that throughout the transition

\[ \frac{2}{m}\leq\xi\leq 1, \qquad \eta\geq-\frac{1}{8}\xi^2. \]

The resulting multipole tilt function is

\[ \class{key-formula-math}{G_{\mathbf p}(\nu) =\left( 1-\nu_1^2-\nu_2^2-\nu_3^2 +k\prod_{j=1}^m\phi_T\bigl(1-\ip{\nu}{p_j}\bigr) \right)^{1/2}.} \]

To guard against floating-point error in the very small sampled minima, all entries in the last column were computed using 90-digit arithmetic.

Pole configuration	$m$	$T$	$k$	Sampled $\min(W-s)$
Coordinate triple	3	$1.35\times10^{-4}$	$5.53\times10^{-5}$	$4.52\times10^{-5}$ ✓
Tetrahedral four	4	$2.47\times10^{-6}$	$1.20\times10^{-8}$	$5.24\times10^{-6}$ ✓
Five-point cluster	5	$3.90\times10^{-11}$	$4.10\times10^{-15}$	$1.17\times10^{-16}$ ✓
Five points with a close pair	5	$1.77\times10^{-12}$	$6.40\times10^{-16}$	$6.39\times10^{-11}$ ✓
Octahedral six	6	$8.28\times10^{-10}$	$1.20\times10^{-18}$	$2.78\times10^{-9}$ ✓

Table 6.1. Finite-$k$ numerical verification of the final power-tail profile.

All sampled margins were positive. This supported the candidate without proving a uniform lower bound, so the rigorous proof used three regions.

Near a zero, freeze the nonvanishing factors and use the direct linear one-pole calculation.
Away from the zeros, use the strict $k\downarrow0$ gap from the previous section.
In a transition annulus, rescale and use the logarithmic ODE; this is laborious but no longer exploratory.

Fix $T$ first to control the pole caps and transition errors, then choose $kT^{2/m-1}\ll1$.

Area-Minimizing Hypersurfaces and Calibration

Mon, 29 Jun 2026 00:00:00 +0000

Definition 1.6.1

A (complete) hypersurface $\Sigma^n$ in a Riemannian manifold $(M^N, g)$ is called (absolutely) area-minimizing if it minimizes the area functional among all hypersurfaces agreeing with $\Sigma$ outside a compact set. More precisely, $\Sigma$ is area-minimizing if for every compact set $K \subset M$, we have

\[ |\Sigma \cap K| \leq |\Sigma' \cap K| \]

for all hypersurfaces $\Sigma'$ with $\Sigma' \setminus K = \Sigma \setminus K$.

We say $\Sigma$ is area-minimizing in its homology class if we also require $\Sigma'$ to be homologous to $\Sigma$, i.e., $\Sigma-\Sigma' = \partial \Gamma$ for some $(n+1)$-dimensional chain $\Gamma$ in $M$.

One can also define area-minimizing submanifolds in other classes, e.g., homotopy classes.

Proposition 1.6.2

Every area-minimizing hypersurface is minimal, i.e., its mean curvature vector $\vec{H} = 0$.

Proposition 1.6.3

Area-minimizing hypersurfaces are stable. That is, the second variation of the area functional is non-negative for all variations with compact support.

Definition 1.6.4

Let $(M, g)$ be a Riemannian manifold. A $k$-form $\omega \in \Omega^k(M)$ is called a calibration if it satisfies:

Closedness: $d\omega = 0$.
Comass $\leq 1$: For every point $p \in M$ and every unit simple $k$-vector $\xi$ at $p$, we have

\[ \omega_p(\xi) \leq 1. \]

Definition 1.6.5

An oriented $k$-dimensional submanifold $\Sigma^k$ in $M$ is calibrated by a $k$-form $\omega$ if $\iota^*\omega = d\mathcal{H}^k|_\Sigma$, where $\iota: \Sigma \hookrightarrow M$ is the inclusion map and $d\mathcal{H}^k|_\Sigma$ is the volume form on $\Sigma$ induced by the Riemannian metric.

Theorem 1.6.6

If an oriented $k$-dimensional submanifold $\Sigma^k$ is calibrated by a closed $k$-form $\omega$, then $\Sigma$ is area-minimizing in its homology class. More precisely, for any other oriented $k$-dimensional submanifold $\Sigma'$ with $\partial\Sigma' = \partial\Sigma$, we have

\[ |\Sigma| \leq |\Sigma'|. \]

Proof. For simplicity, we assume $\Sigma$ is a compact submanifold with boundary. Let $\Sigma'$ be any $k$-dimensional oriented surface with $\partial\Sigma' = \partial\Sigma$. Define the $(k+1)$-dimensional chain $\Gamma$ such that $\partial\Gamma = \Sigma - \Sigma'$.

So we have

\[ \int_\Sigma \omega - \int_{\Sigma'} \omega = \int_\Gamma d\omega = 0, \]

since $\omega$ is closed. Now, we have

\[ |\Sigma|=\int_{ \Sigma} \omega = \int_{ \Sigma'} \omega \leq |\Sigma'|, \]

which shows that $\Sigma$ minimizes area among all surfaces with the same boundary. ◻

Theorem 1.6.7

Any complex analytic variety (i.e., complex submanifold or more generally, integral current defined by a holomorphic equation) in $\mathbb{C}^2$ is absolutely area-minimizing.

Proof. The proof uses the theory of calibrations. In $\mathbb{C}^2$, consider the standard Kähler form

\[ \omega = \frac{i}{2} (dz_1 \wedge d\bar{z}_1 + dz_2 \wedge d\bar{z}_2)=dx_1 \wedge dy_1 + dx_2 \wedge dy_2, \]

The real $2$-form $\omega$ is closed ($d\omega = 0$) and has comass $1$, i.e., for any oriented $2$-plane $\xi$ in $\mathbb{C}^2$, $\omega|_\xi \leq 1$ with equality if and only if $\xi$ is a complex line.

Any complex curve (complex $1$-dimensional submanifold) $\Sigma \subset \mathbb{C}^2$ is calibrated by $\omega$, since the restriction of $\omega$ to $\Sigma$ is exactly the area form of $\Sigma$:

\[ \omega|_\Sigma = d\mathcal{H}^2|_\Sigma. \]

By the preceding calibration argument, $\Sigma$ is area-minimizing in its homology class. Also note that $\mathbb{R}^4$ has trivial second homology group, so any two surfaces with the same boundary are automatically homologous.

Therefore, any complex analytic variety in $\mathbb{C}^2$ is absolutely area-minimizing. ◻

The set $\{ z^{2}=w^3 \}$ is a complex analytic variety in $\mathbb{C}^2$ with an isolated singularity at the origin. It is area-minimizing, but not smooth.

7. Summary and outlook

Thu, 27 Aug 2026 00:00:00 +0000

The construction developed through four successive ideas: the capillary prototype suggested a coercive background plus a small detector; numerical screening isolated the balanced exponent $2/m$; the one-pole calculation forced a linear model at each zero; and the logarithmic ODE supplied the transition between these two regimes. The computations guided the search, while the final conclusions still required rigorous mathematical proof.

AI-assisted symbolic and numerical tools greatly accelerated the testing, comparison, and correction of candidate formulas. We expect increasingly capable AI systems to contribute even more to mathematical research—not by replacing mathematical understanding, but by helping us explore ideas more quickly and deepen that understanding.

Minimal Graphs

Mon, 29 Jun 2026 00:00:00 +0000

Definition 1.7.1

Let $\Omega \subset \mathbb{R}^n$ be an open domain and $u: \Omega \to \mathbb{R}$ be a smooth function. The graph of $u$ is the submanifold

\[ \Sigma_u = \{(x, u(x)) : x \in \Omega\} \subset \mathbb{R}^{n+1}. \]

We say $\Sigma_u$ is a minimal graph if it is a minimal hypersurface in $\mathbb{R}^{n+1}$, i.e., its mean curvature vanishes identically.

The graph $\Sigma_u$ is parametrized by the immersion $\iota: \Omega \to \mathbb{R}^{n+1}$ given by $\iota(x) = (x, u(x))$. The tangent vectors are

\[ \eta_i=\partial_i \iota = \partial_i + \partial_i u \, \partial_{n+1}, \quad i = 1, \ldots, n, \]

where $\{\partial_1, \ldots, \partial_{n+1}\}$ is the standard basis of $\mathbb{R}^{n+1}$.

Basic geometry of graphs.

The induced metric on $\Sigma_u$ is given by

\[ g_{ij} = \delta_{ij} + \partial_i u \, \partial_j u, \]

and its determinant is

\[ \det(g_{ij}) = 1 + |\nabla u|^2, \]

where $|\nabla u|^2 = \sum_{i=1}^n (\partial_i u)^2$. Thus, the area functional of the graph over $\Omega$ is

\[ \mathcal{A}(u) = \int_\Omega \sqrt{1 + |\nabla u|^2} \, dx. \]

The upward-pointing unit normal to $\Sigma_u$ is

\[ \nu = \frac{(-\nabla u, 1)}{\sqrt{1 + |\nabla u|^2}} = \frac{1}{W}(-\partial_1 u, \ldots, -\partial_n u, 1), \]

where we denote $W = \sqrt{1 + |\nabla u|^2}$.

The second fundamental form of $\Sigma_u$ is computed by

\[ A_{ij} = -\left\langle \partial_i \nu, \partial_j \iota \right\rangle = \frac{\partial_i \partial_j u}{W} = \frac{u_{ij}}{W}. \]

The inverse of the induced metric is

\[ g^{ij} = \delta^{ij} - \frac{\partial_i u \, \partial_j u}{W^2}. \]

Proposition 1.7.2

The mean curvature of the graph $\Sigma_u$ is given by

\[ H = \mathrm{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) = \sum_{i=1}^{n} \partial_i \left(\frac{\partial_i u}{\sqrt{1+|\nabla u|^2}}\right). \]

Proof. The mean curvature is the trace of the second fundamental form with respect to the induced metric:

\[ \begin{aligned} H = g^{ij} A_{ij} &= \left(\delta^{ij} - \frac{\partial_i u \, \partial_j u}{W^2}\right)\frac{u_{ij}}{W} \\ &= \frac{1}{W}\left(\Delta u - \frac{\partial_i u \, \partial_j u \, u_{ij}}{W^2}\right)\\ &= \frac{1}{W}\Delta u - \frac{\partial_i u \, \partial_j u \, u_{ij}}{W^3}=\mathrm{div}\left(\frac{\nabla u}{W}\right). \end{aligned} \]

◻

Definition 1.7.3

The minimal surface equation (MSE) is the quasilinear elliptic PDE

\[ \mathrm{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) = 0, \]

or equivalently,

\[ (1+|\nabla u|^2)\Delta u - \sum_{i,j=1}^n \partial_i u \, \partial_j u \, \partial_i \partial_j u = 0. \]

The minimal surface equation is the Euler–Lagrange equation of the area functional

\[ \mathcal{A}(u)=\int_\Omega \sqrt{1+|\nabla u|^2}\,dx. \]

Indeed, for any compactly supported variation $\phi \in C_c^\infty(\Omega)$,

\[ \frac{d}{dt}\bigg|_{t=0} \mathcal{A}(u+t\phi) = \int_\Omega \frac{\nabla u \cdot \nabla \phi}{\sqrt{1+|\nabla u|^2}} \, dx = -\int_\Omega \mathrm{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)\phi \, dx. \]

Hence, $u$ is a critical point of $\mathcal{A}$ if and only if it satisfies the minimal surface equation.

Scherk’s surface. In $\mathbb{R}^3$, the function

\[ u(x_1, x_2) = \log\left(\frac{\cos x_1}{\cos x_2}\right) \]

defined on $\Omega = \{|x_1| \lt{} \pi/2\} \cap \{|x_2| \lt{} \pi/2\}$ is a solution of the minimal surface equation, known as Scherk’s first surface.

Catenoid. The catenoid is a minimal surface of revolution in $\mathbb{R}^3$ that can be locally written as a graph $u(r) = \cosh^{-1}(r)$ for $r \geq 1$.

The Dirichlet Problem for the Minimal Surface Equation

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Definition 1.8.1

The Dirichlet problem for the minimal surface equation asks: given a bounded domain $\Omega \subset \mathbb{R}^n$ and boundary data $\phi \in C^0(\partial\Omega)$, find $u \in C^2(\Omega) \cap C^0(\bar{\Omega})$ such that

\[ \begin{cases} \mathrm{div}\left(\dfrac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) = 0 & \text{in } \Omega, \\ u = \phi & \text{on } \partial\Omega. \end{cases} \]

Definition 1.8.2

A bounded $C^2$ domain $\Omega \subset \mathbb{R}^n$ is mean convex if the mean-curvature vector of $\partial\Omega$ points weakly into $\Omega$. Equivalently, with the scalar convention for which Euclidean balls are mean convex, the boundary mean curvature satisfies $H_{\partial\Omega} \geq 0$.

Theorem 1.8.3

Let $\Omega \subset \mathbb{R}^n$ be a bounded $C^2$ domain. Then the Dirichlet problem for the minimal surface equation has a solution $u \in C^2(\Omega) \cap C^0(\bar{\Omega})$ for every boundary data $\phi \in C^0(\partial\Omega)$ if and only if $\Omega$ is mean convex. In this case the solution is unique. If, moreover, $\partial\Omega$ and $\phi$ are $C^{2,\alpha}$, then the solution is $C^{2,\alpha}$ up to the boundary by the standard boundary regularity theory for quasilinear elliptic equations.

Thus mean convexity is part of the existence theorem, not merely a technical regularity assumption. On a non-mean-convex bounded domain, arbitrary boundary data need not be solvable; Jenkins–Serrin instead prove solvability under an additional smallness condition involving $\operatorname{osc}(\phi)$ and the first two boundary derivatives of $\phi$.

Definition 1.8.4

For a minimal graph $\Sigma_u = \{(x, u(x)) : x \in \Omega\}$, the direction field is defined as

\[ X = \frac{(-\nabla u, 1)}{\sqrt{1 + |\nabla u|^2}}. \]

This direction field satisfies

\[ \begin{aligned} \mathrm{div}\, X &= 0, \\ X \cdot \nu &= 1. \end{aligned} \]

where $\nu$ is the unit normal to $\Sigma_u$.

Theorem 1.8.5

Every minimal graph in $\mathbb{R}^{n+1}$ is area-minimizing within the class of hypersurfaces with the same boundary.

Proof. Define the vector field

\[ V=\frac{(-\nabla u, 1)}{\sqrt{1 + |\nabla u|^2}}. \]

Then the form $\omega = \iota_V d\mathrm{vol}$ is a calibration, and $\Sigma_u$ is calibrated by $\omega$. Therefore, $\Sigma_u$ minimizes area among all hypersurfaces with the same boundary. ◻

Corollary 1.8.6

Every minimal graph in $\mathbb{R}^{n+1}$ is stable.

Bernstein’s Theorem and Generalizations

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Theorem 1.9.1

Let $u: \mathbb{R}^2 \to \mathbb{R}$ be an entire solution of the minimal surface equation. Then $u$ is an affine function, i.e., $u(x_1,x_2) = ax_1 + bx_2 + c$ for some constants $a, b, c \in \mathbb{R}$.

This result was generalized to higher dimensions:

Theorem 1.9.2

Let $u: \mathbb{R}^n \to \mathbb{R}$ be an entire solution of the minimal surface equation.

If $n \leq 7$, then $u$ must be affine.
For $n \geq 8$, there exist non-affine entire solutions (as shown by a counterexample due to Bombieri–De Giorgi–Giusti [BDGG69]).

Proof. Let

\[ \Sigma=\{(x,u(x)):x\in \mathbb{R}^2\}\subset \mathbb{R}^3 \]

be the graph of an entire solution of the minimal surface equation. By the calibration argument above, $\Sigma$ is area-minimizing. In particular, $\Sigma$ is stable, so for every $\varphi\in C_c^\infty(\Sigma)$,

\[ \int_\Sigma |A|^2\varphi^2\,d\Sigma \leq \int_\Sigma |\nabla^\Sigma \varphi|^2\,d\Sigma, \]

since $\mathrm{Ric}_{\mathbb{R}^3}=0$.

We first establish a quadratic area bound. For a.e. $R\gt{}0$, the intersection

\[ \Gamma_R:=\Sigma\cap \partial B_R \]

is a smooth $1$-cycle in the sphere $\partial B_R=S_R$. Since $H_1(S_R)=0$, the curve $\Gamma_R$ bounds a region $D_R\subset S_R$. Replacing $D_R$ by its complement if necessary, we may assume

\[ |D_R|\leq \frac{1}{2}|S_R|=2\pi R^2. \]

Since $\Sigma$ is area-minimizing and $\partial(\Sigma\cap B_R)=\Gamma_R=\partial D_R$, we obtain

\[ |\Sigma\cap B_R|\leq |D_R|\leq 2\pi R^2 \]

for a.e. $R\gt{}0$.

Now choose the logarithmic cutoff

\[ \eta_R(x)= \begin{cases} 1, & |x|\leq R,\\[4pt] \dfrac{k+\log R-\log |x|}{k}, & R\lt{}|x|\lt{}e^{k}R,\\[8pt] 0, & |x|\geq e^{k}R, \end{cases} \]

Here $|x|$ denotes the Euclidean distance to the origin in $\mathbb{R}^3$. Since $|\nabla^\Sigma |x||\leq 1$, on the annulus $R\lt{}|x|\lt{}e^{k}R$ we have

\[ |\nabla^\Sigma \eta_R|\leq \frac{1}{k|x|}. \]

Applying stability with $\varphi=\eta_R$ gives

\[ \int_\Sigma |A|^2\eta_R^2\,d\Sigma \leq \int_\Sigma |\nabla^\Sigma \eta_R|^2\,d\Sigma. \]

To estimate the right-hand side, decompose

\[ B_{e^{k}R}\setminus B_R=\bigcup_{i=0}^{k-1}\bigl(B_{e^{i+1}R}\setminus B_{e^iR}\bigr), \]

Using the area bound,

\[ \begin{aligned} \int_\Sigma |\nabla^\Sigma \eta_R|^2\,d\Sigma &\leq \frac{1}{k^{2}} \sum_{i=0}^{k-1}\int_{\Sigma\cap (B_{e^{i+1}R}\setminus B_{e^iR})} \frac{1}{|x|^2}\,d\Sigma \\ &\leq \frac{1}{k^2} \sum_{i=0}^{k-1}\frac{1}{(e^iR)^2}\, |\Sigma\cap B_{e^{i+1}R}| \\ &\leq \frac{1}{k^2} \sum_{i=0}^{k-1}\frac{1}{(e^iR)^2}\,2\pi (e^{i+1}R)^2 \\ &\leq \frac{C}{k}. \end{aligned} \]

Hence

\[ \int_{\Sigma\cap B_R} |A|^2\,d\Sigma \leq \frac{C}{k}. \]

Letting $k\to \infty$, we conclude that $A\equiv 0$ on $\Sigma \cap B_R$. Since this holds for a.e. $R\gt{}0$, we have $A\equiv 0$ on $\Sigma$. Therefore $\Sigma$ is totally geodesic, hence a plane in $\mathbb{R}^3$.

Since $\Sigma$ is a graph over $\mathbb{R}^2$, that plane cannot be vertical. Thus

\[ u(x_1,x_2)=ax_1+bx_2+c \]

for some constants $a,b,c\in\mathbb{R}$. This proves the theorem. ◻

The key point in dimension $2$ is that stability plus the quadratic area growth

\[ |\Sigma\cap B_R|\leq C R^2 \]

allows the logarithmic cutoff to force

\[ \int_\Sigma |A|^2\,d\Sigma=0. \]

This argument is specific to two dimensions and yields the classical stability proof of Bernstein’s theorem.

Preface

Mon, 29 Jun 2026 00:00:00 +0000

These notes are based on a course on minimal hypersurfaces given at the Beijing Institute of Mathematical Sciences and Applications (BIMSA). The goal is to give a concise introduction to several central ideas in the theory of minimal hypersurfaces, with an emphasis on stability, curvature estimates, regularity, and applications to scalar curvature and mathematical general relativity.

The material begins with basic Riemannian geometry and the first and second variation formulas. It then discusses geometric measure theory, regularity and compactness theorems for stable minimal hypersurfaces, stable Bernstein-type problems, and recent applications of minimal hypersurface methods to positive scalar curvature, the positive mass theorem, and related rigidity questions. The exposition is intended to be self-contained enough for graduate students in geometry, while still keeping the main analytic and geometric mechanisms visible.

Comments, corrections, and suggestions are welcome and may be sent to wanggaoming@bimsa.cn.

References

Mon, 29 Jun 2026 00:00:00 +0000

References

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Scalar-mean rigidity theorem and Llarull's theorem for nonspin manifolds

Mon, 14 Sep 2026 17:21:34 +0000

Main theorem

Spherical rigidity without spin

Llarull's theorem in all dimensions

Let $(M^n,g)$ be a smooth, connected, closed Riemannian manifold with $n\geq2$, and let $F:M\to\mathbb S^n$ be a smooth map of nonzero degree to the unit round sphere. If $F$ is $1$-Lipschitz and $\operatorname{Sc}_g\geq n(n-1)$, then $F$ is an isometry. No spin assumption is required.

Proof architecture

The proof reduces spherical rigidity to a weighted scalar-mean comparison problem on complete manifolds, then descends in dimension while controlling the singularities of the capillary hypersurfaces.

Step 1 · Reduction

From the sphere to weighted scalar-mean comparison

Assuming rigidity fails, conformal deformations on a cylinder over $M$ produce a complete manifold with uniformly positive scalar curvature and a strict boundary mean curvature gap above the trace norm of the comparison map. The map retains nonzero degree. A weighted formulation of this comparison provides the curvature conditions needed for dimension descent.

Step 2 · Capillary bubbles

Pass to a hypersurface and obtain a spectral inequality

A barrier potential controls the noncompact end and produces a compact minimizing weighted capillary $\mu$-bubble. Its stability inequality yields a spectral weighted scalar-mean comparison in one lower dimension. The remaining obstruction to iteration is that the bubble may have interior or boundary singularities.

Step 3 · Singularities and completeness

Blow up the singular set while retaining the degree

Assouad dimension estimates give quantitative control of the singular set across scales. They control a Green-function blow-up that sends the singularities to infinity and equips the regular part with a complete smooth conformal metric. The boundary comparison map is modified near the new ends while preserving the nonzero degree and the spectral comparison needed for induction.

Step 4 · Close the dimension descent

Recover pointwise inequalities and reach dimension two

A positive function obtained from the spectral inequality adjusts the weight and restores pointwise weighted scalar and boundary mean curvature inequalities. This reproduces the complete comparison problem in one lower dimension. Iteration reaches a two-dimensional bubble, where Gauss–Bonnet contradicts the retained nonzero degree, completing the comparison theorem and hence Llarull rigidity.

The stable Bernstein theorem in $\mathbb{R}^{7}$

Mon, 14 Sep 2026 15:19:59 +0000

Main theorem

Stable Bernstein theorem

Flatness in $\mathbb R^7$

Every smooth, connected, complete, two-sided stable minimal immersion $F:M^6\to\mathbb R^7$ has $A\equiv0$, and its image is a hyperplane. No properness, volume-growth, or curvature bound is assumed.

Proof architecture

The proof first establishes all subcritical Green-function moments, then uses critical energy identities and the terminal Green flux to force flatness. Write $A$ for the second fundamental form and $G$ for the minimal positive Green function with a unit pole. For a fixed small $t_0>0$, set

$$ J(t)=\int_{{G=t}}|\nabla G|^3,dA_g, \qquad \mathcal M_\sigma=\int_0^{t_0}J(t)t^{-\sigma-1},dt. $$

Part I · All subcritical moments

Step 1 · Preparation

Geometric reduction and Green functions

A blow-up argument reduces the problem to bounded curvature. Unit Green flux and the estimate $|\nabla G|\leq CG$ give the initial integrability range $\mathcal M_\sigma<\infty$ for $\sigma<2$. The target is every $\sigma<5/2$, the critical exponent in dimension six.

Step 2 · Initial improvement

From $2$ to $2.46$

Bochner's formula, Simons' identity, stability, and the divergence-free tensor $\tfrac12|A|^2g-A^2$ supply the integral identities. The full Codazzi constraint sharpens the directional estimates, while a Schoen–Simon–Yau inequality retaining its Laplacian term controls cutoff errors. Moment continuation yields $\mathcal M_{2.46}<\infty$.

Step 3 · Nonlinear stability

From $2.46$ to $2.4962$

Apply stability to a nonlinear norm combining a curvature test function and a Green-gradient test function. Differentiating this norm retains an extra nonnegative gradient term. Together with the Green and curvature identities, it gives coercive estimates on fifteen rational parameter intervals and proves $\mathcal M_{2.4962}<\infty$.

Step 4 · Spectral coercivity

Every exponent below $5/2$

Additional divergence identities for homogeneous curvature tensors involving $A^4$, $\operatorname{tr}(A^3)A$, and $\operatorname{tr}(A^4)g$ strengthen the coercive estimate near the critical exponent. They continue the moment range from $2.4962$ to every $\sigma<5/2$.

Part II · Critical energies and rigidity

Step 5 · Critical energies

Remove the exterior cutoffs

All subcritical moments are now finite. An elementary spectral inequality, stability, and Simons' identity give finite critical curvature and derivative energies. The argument uses each fixed subcritical moment, with no uniform bound required as $\sigma\uparrow5/2$.

Step 6 · Rigidity

Terminal flux forces flatness

The energy bounds give finite limits of the normalized Green flux $t^{-5/2}J(t)$ at both ends. Brendle's sharp isoperimetric inequality determines the sign of the terminal Bochner boundary term. Combining this sign with the critical spectral inequality forces the critical curvature energy to vanish, so $A\equiv0$.

Mixed Radial Volume Comparison under Spectral Ricci Bounds

Wed, 02 Sep 2026 00:00:00 +0000

Main results

This paper develops radial volume comparison under a spectral lower bound for Ricci curvature. The mixed radial construction separates the conformal geometry used to choose rays from the weights used to truncate and measure them.

Sharp comparison

A space-form model for mixed radial balls

Under the spectral inequality $\operatorname{Ric}_g-\alpha(\Delta_g u/u)g\geq(n-1)\kappa g$, the weighted volume of the $g$-arclength mixed radial domain satisfies a scale-invariant comparison with the corresponding space-form volume.

Polynomial growth

No pointwise bounds on the spectral weight

For an explicit range of radial and measure exponents, mixed radial balls have polynomial weighted-volume growth. The optimal exponent can attain the Euclidean value $n$.

Applications

Diameter, sharp volume, and stable Bernstein

The comparison gives radial proofs of the spectral Bonnet--Myers and sharp volume theorem of Antonelli--Xu, together with a short volume-growth proof that every complete two-sided stable minimal immersion $M^3\to\mathbb R^4$ is flat.

Proof architecture

The argument begins with an exact Riccati identity for the weighted radial Jacobian along geodesics of the conformal metric $h_\alpha=u^{2\alpha}g$. Taking the $(n-1)$-st root of this Jacobian converts the identity into a one-dimensional Sturm comparison and yields the space-form estimate. For general mixed radial balls, the logarithmic derivatives of the Jacobian and the spectral weight are instead treated as coupled variables; a quadratic estimate and a Bellman-type correction produce the polynomial-growth monotonicity formula. The positive-curvature case also gives an optimal diameter bound among smooth closed examples.

Stability of the Riemannian positive mass theorem in all dimensions

Tue, 01 Sep 2026 17:05:50 +0000

Main result

Positive mass stability

Euclidean convergence after excision

For smooth, connected, complete, oriented manifolds of dimension $n\geq3$, without boundary and with finitely many asymptotically flat ends, assume nonnegative integrable scalar curvature and ADM masses tending to zero along distinguished ends. There are exterior regions containing these ends whose boundary areas tend to zero and which converge to Euclidean space in the pointed measured Gromov–Hausdorff topology, using intrinsic length metrics and arbitrary base points in the exterior regions. No spin assumption is required.

Proof architecture

The argument combines minimal graph coordinates with scalar conformal equations to convert small ADM mass into geometric control of exterior regions.

Step 1 · Coordinates

Construct global minimal graphs

Global minimal graph functions define coordinates asymptotic to the identity at the distinguished end. Conformal changes of the associated graph metrics relate scalar curvature to the Hessians of these functions.

Step 2 · Energy estimates

Control coordinate defects by mass

Scalar solutions of the conformal Laplace equation, together with the qualitative positive mass inequality, give weighted Hessian energy estimates in terms of the original ADM mass.

Step 3 · Excision

Recover Euclidean geometry outside small boundaries

Level selection produces exterior regions with small boundary area and small metric distortion. A Euclidean excision argument then yields pointed measured Gromov–Hausdorff convergence with respect to the intrinsic metrics.

Stable Minimal Hypersurfaces in Positively Curved $4$-Manifolds

Sun, 23 Aug 2026 07:45:35 +0000

Main results

This paper studies complete two-sided stable minimal hypersurfaces of dimension three in positively curved four-manifolds. It identifies the curvature gap that forces rigidity and shows, by an explicit construction, why that gap is essential.

Rigidity

Positive scalar curvature forces total geodesicity

If the ambient sectional curvature is nonnegative and the ambient scalar curvature satisfies $\overline R\geq\kappa>0$, then every complete, connected, two-sided stable minimal immersion $M^3\to X^4$ has $A\equiv0$ and $\overline{\operatorname{Ric}}(\nu,\nu)\equiv0$.

Sharpness

The uniform scalar curvature gap is necessary

There is a complete metric with strictly positive sectional curvature on $\mathbb R^4$ containing a complete, embedded, one-ended, nonparabolic, two-sided stable minimal hypersurface $\Sigma^3\cong\mathbb R^3$ that is not totally geodesic.

Improvement

No bounded-geometry or upper-curvature hypothesis

The rigidity theorem removes the weakly bounded geometry assumption from the earlier result of Chodosh--Li--Stryker. It also requires no uniform upper bound for the ambient sectional curvature.

Proof architecture

The argument replaces extrinsic volume comparison by an intrinsic spectral and potential-theoretic route.

Step 1

Spectral and topological reduction

A positive Jacobi function turns stability into spectral Ricci and scalar curvature inequalities. Sharp splitting and topology theorems reduce the only nontrivial case to a one-ended manifold diffeomorphic to $\mathbb R^3$.

Step 2

Separating warped $\mu$-bubbles

Escaping warped $\mu$-bubble separators are constructed with a uniform integral estimate for their mean curvature, the prescribed curvature term, and the normal derivative of the Jacobi weight.

Step 3

Harmonic functions and flux

A level-set identity gives a uniform $L^1$ bound for capacitor gradients. Nonparabolicity would contradict conservation of flux, so the hypersurface is parabolic; stability cutoffs then force the Jacobi potential to vanish identically.

The counterexample mechanism

Starting from the Chodosh–Li–Stryker positively curved model, the ambient metric is changed only near a compact subset of a totally geodesic stable hypersurface. The perturbation introduces a nonzero trace-free second fundamental form while preserving both the induced metric and the Jacobi potential

$$ |A|^2+\overline{\operatorname{Ric}}(\nu,\nu). $$

Thus completeness, minimality, and stability remain unchanged, while the hypersurface ceases to be totally geodesic. Because the deformation is compactly supported and small in $C^2$, strict positivity of sectional curvature is preserved.

Regularity of branched stable minimal immersed hypersurfaces

Fri, 31 Jul 2026 17:09:37 +0000

Main results

This work develops a regularity theory for branched stable minimal immersed hypersurfaces. The conclusions are sharp, and the examples show that the remaining singularities are a genuine feature of the immersed setting rather than an artifact of the proof.

Sharp regularity

An optimal bound for non-branch singularities

The non-branch singular set is empty in dimension $n=2$, discrete in dimension $n=3$, and has Hausdorff dimension at most $n-3$ for every $n\geq4$.

Sharpness examples

A stable cone with a genuine non-branch singularity

We construct a non-flat stable minimal cone in $\mathbb R^4$ arising from a branched minimal immersion. Its products with Euclidean factors attain the $n-3$ bound in every dimension $n\geq3$.

Analytic mechanism

A branched sheeting theory

The proof combines a generalized Schoen inequality with a sheeting theorem near stationary classical cones and unions of hyperplanes.

Starting model for the construction

The construction of our sharpness examples starts from a minimal surface in $\mathbb S^3$ obtained by desingularizing two orthogonal Clifford tori. The image below visualizes this starting model, which provides the geometric input for the subsequent construction of a branched stable minimal cone in $\mathbb R^4$.

Starting model for the sharpness examples: a numerical visualization of the $m=30$ Choe–Soret surface in $\mathbb S^3$, shown in stereographic projection. This depicts the starting surface, not the final branched stable cone. The bronze finish and pedestal are presentation elements.

Companion materials

The two resources below give a closer view of how the project developed and of the geometry behind one of its principal examples.

Research record

How we found the test function

This note follows the route from the capillary prototype through numerical screening, the small-$k$ expansion, the local linear repair, and the cutoff ODE. GitHub Copilot and Cursor Composer 2 were used chiefly to turn hand calculations into rapidly testable code; the numerical evidence guided the choice of formula, while the final verification remained analytic.

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Interactive geometry

Desingularization of two orthogonal Clifford tori

An interactive 3D viewer provides a geometric preview of the desingularization: rotate and zoom the model, or isolate individual regions to inspect how the two tori are joined.

Open the 3D viewer

Regularity of stable capillary minimal hypersurfaces

Wed, 20 May 2026 09:49:07 +0000

Main results

This work gives a regularity and compactness theory for stable capillary minimal hypersurfaces for every contact angle $\theta\in(0,\pi)$.

Regularity and compactness

Angle-dependent sharp dimension bounds

The singular set is empty below dimension $n_\theta$, discrete in dimension $n_\theta$, and has dimension at most $n-n_\theta$ above it, where $n_\theta$ equals $7$, $6$, or $5$ according to the distance of $\theta$ from $90^\circ$.

Cone classification

Flatness for every contact angle

For $n=3,4$ and every contact angle $\theta\in(0,\pi)$, a stable capillary minimal hypercone with an isolated singularity is flat.

Bernstein consequence

Flatness for $2\leq n\leq4$

For $2\leq n\leq4$, every properly embedded complete two-sided stable capillary minimal hypersurface in $\mathbb H^{n+1}$ with Euclidean area growth is flat.

Proof architecture

The main difficulty is designing a capillary tilt function that both satisfies a differential Schoen inequality and is compatible with the capillary boundary condition, so that the boundary terms cancel after a carefully weighted integration. Once this inequality is established, the De Giorgi iteration used by Bellettini yields a boundary sheeting theorem, which then leads to the regularity and compactness theory.

Spectral splitting theorem and ends of minimal hypersurfaces

Thu, 14 May 2026 15:06:46 +0000

Main results

The paper develops a geodesic-line approach to spectral Ricci curvature and uses it to control the topology at infinity of minimal hypersurfaces.

Spectral splitting

A weighted minimizing line recovers classical rigidity

If $\lambda_1(-\alpha\Delta+\operatorname{Ric})\geq0$ with $\alpha<4/(n-1)$ and the associated weighted metric admits a minimizing line, then $\operatorname{Ric}\geq0$ pointwise and the manifold splits off an $\mathbb R$ factor.

Ends at infinity

Infinitely many ends create a weighted line

Nonnegative spectral Ricci curvature outside a compact set, together with infinitely many ends, forces the existence of a weighted minimizing geodesic line outside a larger compact set.

Minimal hypersurfaces

Finite index implies finitely many ends

Every complete two-sided finite-index minimal hypersurface in an ambient manifold of dimension at most five with nonnegative biRic curvature has finitely many ends, without a properness assumption.

Proof strategy

The splitting proof adapts the Cheeger–Gromoll Busemann-function argument to the weighted length functional determined by a positive solution of the spectral equation. For the application, infinitely many ends are used to construct a weighted line at infinity; the positive solution is then perturbed to make the spectral inequality strict outside a compact set. The second variation of weighted length produces the contradiction and hence finiteness of ends.

A splitting theorem for manifolds with spectral nonnegative Ricci curvature and mean-convex boundary

Fri, 23 Jan 2026 11:51:14 +0800

Main results

This paper proves a boundary splitting theorem under a spectral, rather than pointwise, lower bound for Ricci curvature.

Spectral splitting

An interior end forces a half-product

Let $M^n$ have mean-convex boundary and $\lambda_1(-\alpha\Delta+\operatorname{Ric})\geq0$ for $\alpha<4/(n-1)$. If $M$ has an interior end, then it is isometric to $\Sigma\times\mathbb R_{\geq0}$ with $\Sigma$ closed and $\operatorname{Ric}_\Sigma\geq0$.

Dichotomy

Otherwise there are no interior ends

The theorem allows a possibly noncompact boundary and gives a sharp alternative: either the product splitting occurs or every end of the manifold remains attached to the boundary.

Geometric application

Stable free-boundary CMC hypersurfaces inherit the dichotomy

In dimensions $n\leq4$, suitable nonnegative biRic curvature and mean convexity imply the same product-or-no-interior-end conclusion for complete two-sided stable free-boundary constant-mean-curvature immersions.

Proof strategy

A positive solution of the spectral Ricci equation defines the weighted length functional $L_u^\alpha$. An interior end produces a free-boundary weighted minimizing ray, and a curve-capture argument constructs such a ray or line through every point. Its second variation forces the weight to be constant and hence upgrades spectral nonnegativity to pointwise $\operatorname{Ric}\geq0$; the classical boundary splitting theorem then finishes the proof.

Scalar curvature rigidity of domains in a 3-dimensional warped product

Thu, 06 Mar 2025 11:51:14 +0800

Main results

The paper establishes Llarull-type scalar-curvature rigidity for domains in three-dimensional spherical warped products, including models with boundary and singular endpoint geometry.

Domain rigidity

Metric, scalar, and boundary comparison determine the model

For a domain whose side boundary is convex in the conformally related product metric, the inequalities $g\geq\bar g$, $R_g\geq R_{\bar g}$, and $H_g\geq H_{\bar g}$ force $g=\bar g$ under the stated endpoint hypotheses.

Endpoint geometry

Smooth, conical, and disk-type ends are treated uniformly

The rigidity theorem covers smooth poles, Euclidean or metric tangent cones, and truncations by end disks with matching mean-curvature and angle inequalities.

Conical Llarull theorem

Antipodal cone points are allowed

For warped three-spheres with $\psi(t_\pm)=a_\pm|t-t_\pm|+o(|t-t_\pm|)$ and $0

Proof strategy

Prescribed-mean-curvature capillary $\mu$-bubbles play the role of the minimizing hypersurfaces in Llarull’s theorem. The stability inequality and a Gauss–Bonnet comparison make a minimizing bubble infinitesimally rigid, and a local foliation propagates equality through the domain. Near a vanishing warping factor or a conical endpoint, local barriers and tangent-cone analysis replace a formal singular perturbation and ensure that the minimizing problem cannot escape through the pole.

A splitting theorem for 3-manifold with nonnegative scalar curvature and mean-convex boundary

Thu, 16 Jan 2025 04:51:14 +0800

Main results

This work proves scalar-curvature splitting theorems for noncompact three-manifolds with mean-convex boundary, using noncompact minimizing free-boundary and capillary surfaces as the rigidifying objects.

Free-boundary splitting

A minimizing half-cylinder or strip forces flatness

If the manifold contains an absolutely area-minimizing free-boundary half-cylinder or strip, then it is flat. Up to scaling, the resulting models are covered by $\mathbb S^1\times\mathbb R^2_+$ or are isometric to $[0,1]\times\mathbb R^2$.

Capillary extension

The theorem holds for every contact angle

For $\theta\in(0,\pi)$, an absolutely $\theta$-energy-minimizing half-cylinder gives the corresponding tilted flat model; the strip statement also holds when its two boundary curves lie on different boundary components.

No compact-boundary hypothesis

The ambient boundary may be noncompact or disconnected

The argument requires neither compactness nor a prescribed number of components of $\partial M$, extending earlier boundary splitting theorems beyond their usual global assumptions.

Proof strategy

Plateau-type problems in carefully chosen prism-shaped domains produce a sequence of minimizing capillary surfaces near the given noncompact surface. Stability classifies the possible limits, while energy comparisons rule out unwanted disk, half-plane, plane, and cylinder degenerations. The surviving half-cylinders or strips form a rigid family; equality in the scalar-curvature and boundary inequalities then propagates to an ambient flat product.

Index of Embedded Networks in the Sphere

Tue, 01 Oct 2024 11:51:01 +0800

Main results

This paper develops a spectral decomposition method for stationary geodesic networks and applies it to every embedded closed triple-junction network in the round two-sphere.

Exact index

The Morse index is determined by the complementary regions

If the network cuts $\mathbb S^2$ into $F$ regions, then its Morse index is exactly $F-1$. All negative modes have eigenvalue $-1$ and are locally constant along the edges.

Exact nullity

The only Jacobi fields come from rotations

Every embedded closed stationary triple-junction network has nullity three, with the zero modes generated precisely by infinitesimal rotations of $\mathbb S^2$.

Decomposition formula

Large networks reduce to smaller pieces

For an arbitrary partitioned network, the index and nullity split into contributions from the subnetworks, a finite-dimensional Dirichlet-to-Neumann operator, and explicit matching spaces at the cut points.

Method

The Jacobi operator is $L=\Delta+1$, acting on edgewise functions satisfying the compatibility conditions imposed by geometric variations at junctions. After cutting the network, solutions of the Jacobi equation on each piece are encoded by a self-adjoint Dirichlet-to-Neumann map. The resulting index formula turns the global spectral calculation into finite-dimensional linear algebra and, together with the classification of spherical triple-junction networks, yields the exact spectrum at and below zero.

On $\delta$-Stable Minimal Hypersurfaces in $\mathbb{R}^{n+1}$

Thu, 04 Jul 2024 11:51:14 +0800

Main results

The paper extends regularity, compactness, and Bernstein theory from stable minimal hypersurfaces to the weaker notion of $\delta$-stability.

Regularity and compactness

A quantitative theory above the catenoid threshold

For $n\geq3$ and $\delta>(n-2)/n$, sequences with uniform area control admit varifold subsequences whose singular sets satisfy an explicit dimension bound determined by $\delta$.

Bernstein theorem

Flatness in dimensions three and four

For $n=3,4$, complete two-sided $\delta$-stable immersed hypersurfaces are flat under either a stronger numerical stability range or the combination of simple connectivity, properness, and finitely many ends.

Sharp threshold

The lower bound on $\delta$ cannot be improved

The $n$-dimensional catenoid is exactly $(n-2)/n$-stable. Simons-type cones also identify the sharp threshold for the cone-classification component of the theory.

Proof strategy

The regularity argument combines a Caccioppoli inequality, De Giorgi iteration, an $\varepsilon$-regularity theorem for the second fundamental form, and tangent-cone analysis. A sharp classification of $\delta$-stable minimal cones supplies the dimension reduction. In low dimensions, point-picking and curvature estimates reduce the Bernstein statements to the global flatness results.

Allard-Type Regularity for Varifolds with Prescribed Contact Angle

Tue, 26 Mar 2024 11:51:14 +0800

Main results

This paper establishes a boundary Allard theorem for varifolds with prescribed contact angle, including variable angles and ambient metrics of low regularity.

Boundary regularity

A multiplicity-one tangent half-plane gives smooth structure

If a boundary point admits a multiplicity-one tangent half-hyperplane, then the support is locally a $C^{1,\gamma}$ hypersurface with boundary for some $\gamma\in(0,1)$.

Contact angle

The variational angle is recovered geometrically

On the regular support, the geometric contact angle is the prescribed angle (apart from the explicitly identified orthogonal alternative); the ambiguity disappears under a natural nonorthogonality condition.

Density criterion

Mass control alone suffices in favorable regimes

For two-dimensional varifolds, or in every dimension when the angle is sufficiently close to $90^\circ$, a quantitative boundary density bound implies the required regularity.

Proof strategy

The proof is carried out in a flattened half-space equipped with a $C^1$ metric, so it remains intrinsic to the codimension-one contact-angle problem. It combines a boundary monotonicity formula with an $L^2$ height and tilt estimate, a blow-up analysis, and geometric excess decay. A reflection idea inspired by Leon Simon explains how the boundary term and the capillary angle are incorporated into a stationary comparison object.

Dihedral rigidity in hyperbolic 3-space

Wed, 08 Nov 2023 11:51:14 +0800

Main results

The paper proves hyperbolic analogues of Gromov’s dihedral comparison and rigidity conjectures for tetrahedra and a broad class of prisms.

Comparison

All inequalities cannot be simultaneously strict

For the prescribed tetrahedra and prisms, scalar curvature greater than $-6$ and face mean curvature above the hyperbolic reference prevent every dihedral angle from being strictly smaller than its model value.

Rigidity

Weak comparison forces the hyperbolic model

Under the cylinder-trapping condition, or the stated side-angle alternative, the inequalities $R_g\geq-6$, $H_g\geq H_{\mathrm{hyp}}$, and $\gamma_g\leq\gamma_{\mathrm{hyp}}$ imply that the polyhedron is isometric to one in hyperbolic space.

Geometric scope

Every hyperbolic tetrahedron in the class is covered

All cone-type references satisfy cylinder trapping; for prisms it is equivalent to a concrete nesting condition between the two Euclidean base polygons.

Proof strategy

A variational functional produces a stable capillary surface of constant mean curvature $\pm2$ with the contact angles of the reference foliation. Its second variation, together with Gauss–Bonnet, converts scalar-, mean-, and dihedral-curvature comparisons into a contradiction unless equality holds. In the equality case, an infinitesimally rigid capillary surface is propagated by a constant-mean-curvature foliation; a separate foliation construction at a vertex completes the global rigidity argument.

Stable Capillary Minimal Surfaces

Wed, 24 May 2023 00:00:00 +0000

Table of Contents

We set the following notations. Suppose $(M^{n+1},\partial M)$ is the ambient manifold we’re interested in. We say $\Sigma$ is a capillary minimal surface with contact ange $\theta$ if $\Sigma$ is a minimal surface with boundary $\partial \Sigma$ and it has contact angle $\theta$ with $\partial M$ along $\partial \Sigma$.

First Variation Formula

Suppose $\partial \Sigma$ bounds a region $W$ in $\partial M$ with outer unit normal $\overline{\nu}$, then $\Sigma$ is a capillary minimal surface if and only if it is a critical point of the following functional, $$ \mathcal{A}(\Sigma)=\left|\Sigma\right|-\cos \theta \left|W\right|. $$ This can be seen from the following argument.

Let $X$ be the variational vector field. Then, $$ \left.\frac{d}{dt}\right|_{t=0}\mathcal{A}(\Sigma_t)=\int_{\Sigma}\mathrm{div}_\Sigma XdA-\cos\theta\int_{ M}\mathrm{div}_MXdA. $$ Now if we suppose $X|_{\partial M} \in \mathfrak{X}(\partial M)$, then $$ \int_{ M} \mathrm{div}_MXdA= \int_{ \partial \Sigma} X\cdot \overline{\nu} ds. $$

So $$ \left. \frac{d}{dt}\right|_{t=0}\mathcal{A}(\Sigma_t)={} \int_{ \Sigma} X \cdot \nu H dA+\int_{ \partial \Sigma} (X \cdot \eta-\cos \theta X\cdot \overline{\nu})ds. $$ Suppose $\eta$ makes angle $\varphi$ with $T(\partial M)$, (i.e., $\eta=\cos \varphi \overline{\nu}+\sin \varphi \overline{\eta}$), then $X\cdot \eta = \cos \theta X \cdot \overline{\nu}$. Hence, $\Sigma$ is a critical point if and only if $H\equiv 0$ and $\varphi=\theta$ along $\partial \Sigma$.

Fix a unit vector $v \in \partial H=\mathbb{R}^{n-1}$. Choose two constant vectors $\nu_1=-\cos \theta \overline{\eta} + \sin \theta v, \nu_2=-\cos \theta \overline{\eta}-\sin \theta v$.

Then we replace $\varphi$ by $\varphi \sqrt{(1-\nu \cdot \nu_1)(1-\nu \cdot \nu_2)}$ in the stability inequality.

Stability inequality, choose $A(X,Y)= \left< \nabla_{X}Y, \nu \right>$. $$\int_{ \Sigma} \left|\nabla \varphi\right|^2-\left|A\right|^2\varphi^2+ \int_{ \partial \Sigma} \cot \theta A(\eta,\eta) \varphi^2\ge 0.$$

Preliminary identities. $$\begin{aligned} \nu={} & -\cos \theta \overline{\eta}+\sin \theta \overline{\nu} \ \eta={} & \sin \theta \overline{\eta}+\cos \theta \overline{\nu} \ 1 -\nu \cdot \nu_1={}& 1-\cos ^2 \theta-\sin ^2\theta \overline{\nu}\cdot v = \sin ^2\theta(1-\overline{\nu}\cdot v) \ 1 -\nu\cdot\nu_2={}& 1-\cos ^2\theta+\sin ^2\theta \overline{\nu}\cdot v=\sin ^2\theta(1+\overline{\nu}\cdot v)\ \frac{\partial }{\partial \eta}(1-\nu \cdot \nu_1)={}& - \frac{\partial }{\partial \eta}\nu \cdot \nu_1 = A(\eta,\eta)\eta \cdot \nu_1=A(\eta,\eta)(-\sin \theta\cos \theta +\sin \theta\cos \theta \overline{\nu}\cdot v) \ ={}& -\sin \theta\cos \theta A(\eta,\eta)(1-\overline{\nu}\cdot v) \ \frac{\partial }{\partial \eta}(1-\nu\cdot\nu_2)={}& -\sin \theta\cos \theta A(\eta,\eta)(1+\overline{\nu}\cdot v) \end{aligned}$$

Now we may insert $\varphi \sqrt{(1-\nu\cdot\nu_1)(1-\nu\cdot\nu_2)}$ in stability inequality. (Let $w = \sqrt{(1-\nu\cdot\nu_1)(1-\nu\cdot\nu_2)}$) $$\begin{aligned} {} & \int_{ \Sigma} \left|A\right|^2(1-\nu\cdot\nu_1)(1-\nu\cdot\nu_2) \varphi^2\ \le{} & \int_{ \Sigma} w^2 \left|\nabla \varphi\right|^2+\varphi^2\left|\nabla w\right|^2+ \frac{1}{2}\int_{ \Sigma} \nabla \varphi^2\cdot \nabla w^2 \ {}&+\int_{ \partial \Sigma} A(\eta,\eta) \cot \theta \varphi^2 \sin ^4 \theta(1-\overline{\nu}\cdot v)(1+\overline{\nu}\cdot v)\ ={} & \int_{ \Sigma} w^2 \left|\nabla \varphi\right|^2+\varphi^2\left|\nabla w\right|^2- \frac{1}{2}\int_{ \Sigma} \varphi^2 \Delta w^2+ \frac{1}{2}\int_{ \partial \Sigma} \varphi^2 \frac{\partial }{\partial \eta}w^2 \ {}&+\int_{ \partial \Sigma} A(\eta,\eta)\varphi^2 \sin^3 \theta \cos \theta(1-\overline{\nu}\cdot v)(1+\overline{\nu}\cdot v) \end{aligned}$$

We compute $\frac{\partial }{\partial \eta}w^2$. $$\begin{aligned} \frac{\partial }{\partial \eta}w^2={} & (1-\nu\cdot\nu_2)\frac{\partial }{\partial \eta}(1-\nu\cdot\nu_1)+(1-\nu\cdot\nu_1)\frac{\partial }{\partial \eta}(1-\nu\cdot\nu_2)\ ={} & \sin ^2 \theta (1+\overline{\nu}\cdot v)\times - \sin \theta \cos \theta A(\eta,\eta)(1-\overline{\nu}\cdot v)\ {}& +\sin ^2\theta (1-\overline{\nu}\cdot v) \times -\sin \theta\cos \theta A(\eta,\eta)(1+\overline{\nu}\cdot v) \ = {}& -2\sin ^3\theta \cos \theta A(\eta,\eta)(1-\overline{\nu}\cdot v)(1+\overline{\nu} \cdot v) \end{aligned}$$

Hence, the boundary term can be cancelled out.

Let’s focus on $\Delta w^2.$ $$\begin{aligned} \Delta w^2={} & -(1-\nu \cdot \nu_2)\Delta(\nu \cdot \nu_1)-(1-\nu \cdot \nu_1)\Delta(\nu \cdot \nu_2)+2(\nabla \nu \cdot \nu_1)\cdot(\nabla \nu\cdot\nu_2) \ ={} & (1-\nu \cdot\nu_2) \left|A\right|^2(\nu \cdot \nu_1)+(1-\nu\cdot\nu_1)\left|A\right|^2(\nu\cdot\nu_2)+(\nabla \nu \cdot \nu_1)\cdot(\nabla \nu\cdot\nu_2)\ ={}& \left|A\right|^2(\nu\cdot\nu_1+\nu\cdot\nu_2-2 \left( \nu\cdot\nu_1 \right)\cdot (\nu\cdot\nu_2))+(\nabla \nu \cdot \nu_1)\cdot(\nabla \nu\cdot\nu_2) \end{aligned}$$

$$\begin{aligned} LHS={} & \int_{ \Sigma} \left|A\right|^2(1-2\cos ^2 \theta + (\nu\cdot\nu_1)(\nu\cdot \nu_2))\varphi^2 \ \int_{ \Sigma} \left|A\right|^2(1-\cos ^2\theta)\varphi^2\le{} & \int_{ \Sigma} w^2 \left|\nabla \varphi\right|^2+\varphi^2 \left|\nabla w\right|^2-{(\nabla \nu \cdot \nu_1)\cdot(\nabla \nu\cdot\nu_2)}\varphi^2. \end{aligned}$$

$\nu\cdot\nu_1+\nu\cdot\nu_2=2\cos ^2 \theta$

Now we note $$\begin{aligned} \left|\nabla w\right|^2 -(\nabla \nu \cdot \nu_1)\cdot (\nabla \nu\cdot\nu_2)={} & \frac{\left|(1-\nu\cdot\nu_1)(\nabla \nu\cdot\nu_2)-(1-\nu\cdot \nu_2)(\nabla \nu\cdot\nu_1)\right|^2}{4w^2} \end{aligned}$$

Some useful formulas $$\left|\nabla \nu \cdot p\right|^2= \sum_{i =1}^{n}h_{1i}^2(\left|p\right|^2-(p\cdot \nu)^2).$$

Note that $$\begin{aligned} \mathrm{I}:={} & \left|\nabla \nu \cdot \left( \left( 1-\nu\cdot\nu_1 \right)\nu_2-(1-\nu \cdot \nu_2)\nu_1 \right)\right|^2 \ ={} & |\nabla \nu \cdot ( (1+\cos \theta \overline{\eta}\cdot\nu-\sin \theta v \cdot\nu)(-\cos \theta \overline{\eta}-\sin \theta v)\ &-(1+\cos \theta \overline{\eta}\cdot \nu-\sin \theta v\cdot \nu)(-\cos \theta\overline{\eta}+\sin \theta v) )|^2\ ={}& 4\sin ^2 \theta\left|\nabla \nu \cdot (-(1+\cos \theta \overline{\eta}\cdot \nu)v+\cos \theta v \cdot\nu \overline{\eta})\right|^2 \end{aligned}$$

Note that $$(-(1+\cos \theta \overline{\eta}\cdot\nu)v+ \cos \theta v \cdot \nu\overline{\eta})\cdot \nu=- v \cdot \nu.$$

Hence $$\begin{aligned} \mathrm{I}={} & 4\sin ^2\theta \sum_{ }^{}h_{1i}^2((1+\cos \theta \eta\cdot \nu)^2+\cos ^2 \theta \left( v \cdot \nu \right)^2- (v\cdot\nu)^2)\ ={} & 4\sin ^2\theta \sum_{ }^{}h_{1i}^2 \left( (1+\cos \theta \eta\cdot\nu)^2-\sin ^2\theta (v\cdot\nu)^2 \right) \end{aligned}$$

On the other hand, we know $$w^2 = (1+\cos \theta \overline{\eta} \cdot \nu-\sin \theta a\cdot \nu)(1+\cos \theta\overline{\eta}\cdot\nu+\sin \theta v\cdot\nu)$$

So we get $$\mathrm{I}=4 \sin ^2\theta \sum h_{1i}^2 w^2.$$

In summary, we get $$\int_{ \Sigma} \sin ^2\theta \left|A\right|^2\le \int_{ \Sigma} w^2 \left|\nabla \varphi\right|^2+\int_{ \Sigma} \sin ^2\theta \sum h_{1i}^2.$$

Using Schoen-Simon’s inequality, we have the following inequality $$\int_{ \Sigma} \left|A\right|^2\varphi^2\le C_\theta \int_{ \Sigma} (1-\nu\cdot\nu_1)(1-\nu\cdot\nu_2)\left|\nabla \varphi\right|^2.$$

First variation formula

Suppose $\partial \Sigma$ bounds a region $W$ in $\partial M$ with outer unit normal $\overline{\nu}$, then $\Sigma$ is a capillary minimal surface if and only if it is a critical point of the following functional. $$\mathcal{A}(\Sigma)=\left|\Sigma\right|-\cos \theta \left|M\right|.$$

::: proof Proof. Let $X$ be the variational vector field, then $$\begin{aligned} \left. \frac{d}{dt}\right|{t=0} & \mathcal{A}(\Sigma_t)= \int{ \Sigma} \mathrm{div}\Sigma X dA - \cos \theta\int{ M} \mathrm{div}_M X dA.

\end{aligned}$$

Now if we suppose $X|{\partial M} \in \mathfrak{X}(\partial M)$, then $$\int{ M} \mathrm{div}MXdA= \int{ \partial \Sigma} X\cdot \overline{\nu} ds$$

So $$\begin{aligned} \left. \frac{d}{dt}\right|{t=0}\mathcal{A}(\Sigma_t)={} & \int{ \Sigma} X \cdot \nu H dA+\int_{ \partial \Sigma} (X \cdot \eta-\cos \theta X\cdot \overline{\nu})ds \end{aligned}$$

Suppose $\eta$ makes angle $\varphi$ with $T(\partial M)$, (i.e. $\eta=\cos \varphi \overline{\nu}+\sin \varphi \overline{\eta}$), then $X\cdot \eta = \cos \theta X \cdot \overline{\nu}$.

Hence, $\Sigma$ is a critical point if and only if $H\equiv 0$ and $\varphi=\theta$ along $\partial \Sigma$. ◻ :::

Relation of capillary minimal surfaces and minimal triple junction surfaces. Considering $\theta=\frac{2\pi}{3}$, we know $\mathcal{A}(\Sigma)=\left|\Sigma\right|+\frac{1}{2}\left|W\right|$. Let $\Sigma’$ be the reflection of $\Sigma$ with respect to $\partial M$ (e.g. suppose $M$ is a standard half space), then $(\Sigma,\Sigma’, W)$ will be a minimal triple junction surfaces and the energy function for these surfaces should be $\mathcal{A}=\left|\Sigma\right|+\left|\Sigma’\right|+\left|M\right|$.

Second variation formula

$$\int_{ \Sigma} \left|\nabla \phi\right|^2-\left|A\right|^2\phi^2+ \int_{ \partial \Sigma} \left( \frac{A _{\partial M}(\overline{\nu},\overline{\nu})}{\sin \theta}+ \cot \theta A(\eta,\eta) \right)\phi^2.$$

We choose the special vector fields as $X = \left< X, \nu \right>\nu +\left< X, \eta \right> \eta= \phi \nu + \cot \theta \phi \eta=\frac{1}{\sin \theta} \phi \overline{\nu}$.

We still write $\phi= X\cdot \nu$. First term. $$\begin{aligned} \left. \frac{d}{dt}\right|{t=0}\int{ \Sigma}X\cdot \nu_t H_t dA_t= {} & \int_{ \Sigma} X\cdot \nu H’ dA + \int_{ \Sigma} H \left. \frac{d}{dt}\right|{t=0}(X\cdot \nu_t)dA_t \end{aligned}$$ We can use $H’ = \Delta{\Sigma}\phi+\left|A\right|^2\phi$. So the interior part is given by this term.

For the boundary term, we need to compute $$\begin{aligned} \left. \frac{d}{dt}\right|{t=0} \int{ \partial \Sigma} (X\cdot \eta_t - \cos \theta X \cdot \overline{\nu}t)ds_t ={} & \int{ \partial \Sigma} \left< X, \eta-\cos \theta \overline{\nu} \right>’ ds. \end{aligned}$$

So we need to compute $\eta’, \overline{\nu}’$.

First, let’s compute $\nu’$. Let $\left{ e_i \right}$ be the orthonormal basis of $T_p \Sigma$. Then $$\begin{aligned} \left< \nu’, e_i \right> ={} & -\left< \nu, e_i’ \right> = -\left< \nu ,D _{e_i}X \right> =-\left< \nu, D {e_i}(\phi \nu + \left< X, e_j \right> e_j) \right> \ ={} & -e_i(\phi)-\left< X, e_j \right> h{ij} \end{aligned}$$

Hence $\nu’ = - \nabla_{\Sigma}\phi-A(X^\top ,e_i )e_i$.

Now $\left< \eta’, \nu \right> =- \left< \eta,\nu’ \right> =\frac{\partial }{\partial \eta}\psi + A(X^T, \eta)$.

We also have $$\begin{aligned} \left< \eta’, \tau_i \right> ={}& -\left< \eta, \tau_i’ \right> = -\left< \eta, D _{\tau_i}X \right> = \phi A(\eta, \tau_i)+\cot \theta \tau_i(\phi)\ \end{aligned}$$

So $$\eta’ = \left( \frac{\partial \phi}{\partial \eta}+A(X^T, \eta) \right)\nu + \phi A(\eta,\tau_i)\tau_i+\cot \theta \nabla_{\partial \Sigma} \phi.$$

Hence $$\begin{aligned} \left< X, \eta’-\cos \theta \overline{\nu}’ \right> ={} & \frac{\phi}{\sin \theta}\left< \overline{\nu}, \eta’-\cos \theta \overline{\nu}’ \right> =\frac{\phi}{\sin \theta} \left< \overline{\nu}, \eta’ \right> \ ={} & \phi \left( \frac{\partial \phi}{\partial \eta} + \cot \theta\phi A(\eta,\eta)\right) \end{aligned}$$

Moreover, $$\left< X’, \eta - \cos \theta \overline{\nu} \right> = \left< \nabla_{X}X, \sin \theta \overline{\eta} \right> = \frac{\phi^2}{\sin\theta} A _{\partial M}(\overline{\nu},\overline{\nu}).$$

Generalized Bernstein Theorem

We know the curvature estimate for stable capillary surfaces will be related to the generalized Bernstein Theorem in half space $H = \left{ x_1\ge 0 \right}$.

Here, we will prove the generalized Bernstein Theorem for capillary surfaces in $H^3$.

::: theorem Theorem 1. If $(\Sigma,\partial \Sigma) \hookrightarrow (H^3, \partial H)$ is a stable capillary minimal surface with quadratic area growth condition. Then $\Sigma$ is flat. :::

The stability inequality becomes $$\int_{ \Sigma} \left|A\right|^2 \phi ^2dA\le \int_{ \Sigma} \left|\nabla \phi\right|^2dA+ \int_{ \partial \Sigma} \cot \theta A(\eta,\eta)\phi^2ds.$$

Dealing with boundary term

Let’s choose $w = 1+\left< \nu, \overline{\eta} \right>\cos \theta$. Replace $\phi$ by $\phi w$ in stability inequality. We note $$\begin{aligned} \int_{ \Sigma} \left|\nabla (\phi w)\right|^2={} & \int_{ \Sigma} w^2 \left|\nabla \phi\right|^2+ \phi^2 \left|\nabla w\right|^2+\frac{1}{2}\int_{ \Sigma} \left< \nabla \phi^2, \nabla w^2 \right> \ ={} & \int_{ \Sigma} w^2 \left|\nabla \phi\right|^2-\phi^2 w \Delta w + \int_{ \partial \Sigma} \phi^2 w \frac{\partial w}{\partial \eta}. \end{aligned}$$

Note that we have $$\Delta w = \cos \theta \Delta \left< \nu ,\overline{\eta} \right> = -\cos \theta \left< h_{ij}e_j, \overline{\eta}\right> _i= -\cos \theta\left|A\right|^2 \left< \nu, \overline{\eta} \right>$$ and $$\frac{\partial w}{\partial \eta}=\cos \theta \left< \frac{\partial \nu}{\partial \eta}, \overline{\eta} \right> =-\cos \theta \sin \theta A(\eta,\eta)=-\cot \theta A(\eta,\eta)w.$$

Here, we’ve used $w=1-\cos ^2\theta=\sin ^2\theta$ along $\Gamma$.

So in summary, we will get $$\int_{ \Sigma} \left|A\right|^2 w\phi^2 dA\le \int_{ \Sigma} \left|\nabla \phi\right|^2w^2dA.$$

So we will get the following inequality $$\int_{ \Sigma} \left|A\right|^2\phi^2dA\le C_\theta \int_{ \Sigma} \left|\nabla \phi\right|^2dA$$

Choose special test functions (logarithmic cutoff function)

We will control the curvature $\left|A\right|$ in the half ball $B_1 \cap H$.

Choose $\phi$ as follows $$\phi= \begin{cases} 1, & \left|x\right|\le 1, \ 1- \frac{\log \left|x\right|}{n}, & 1\le \left|x\right|\le e^n,\ 0, & \left|x\right|\ge e^n. \end{cases}$$

Then, we can find $$\begin{aligned} \int_{ \Sigma \cap B_1} \left|A\right|^2dA\le {} & C \int_{ \Sigma \cap B {e^n}\backslash B_1} \frac{1}{\left|x\right|^2 n^2} dA\le C \sum{i =1}^{n} \int_{ \Sigma \cap B {e^i}\backslash B{e^{i-1}}}\frac{1}{e^{2i-2} n^2}dA \ \le {} & C \sum_{i =1}^{n} \frac{C e^2}{n^2}= \frac{C}{n}. \end{aligned}$$ Now we may choose $n\rightarrow 0$ to get $\left|A\right|\equiv 0$ in $B_1$. The unique continuation of minimal surfaces tells us $\Sigma$ is flat.

Scalar curvature comparison of rotationally symmetric sets

Tue, 25 Apr 2023 11:51:14 +0800

Main results

This paper treats scalar-curvature rigidity for compact three-manifolds whose reference boundary is rotationally symmetric and weakly convex, including smooth, conical, and truncated models.

Euclidean rigidity

Boundary comparison determines the interior

If $R_g\geq0$, the induced boundary metric and mean curvature dominate those of the Euclidean reference, and the nonsmooth dihedral angles are no larger, then the manifold is flat under the stated pole hypotheses.

Endpoint geometries

Disks, cones, and spherical poles are all included

The theorem covers boundaries cut off by planar disks, boundaries with conical tips, and boundaries with smooth spherical poles. Standard balls and right circular cones are important special cases.

Extensions

Hyperbolic and circle-symmetric analogues

The same capillary framework yields corresponding rigidity statements in hyperbolic space and in nonpositively curved spaces with an $\mathbb S^1$ symmetry.

Proof strategy

The argument minimizes a capillary functional whose prescribed contact angle varies along the rotational boundary. Stability and Gauss–Bonnet force the minimizing disk to be infinitesimally rigid, after which a foliation propagates the equality. The main difficulty is ensuring that a nontrivial minimizer exists near a conical or spherical pole; local constant-mean-curvature barrier foliations reduce those cases to the planar-end problem.

Generalized Bernstein Theorem for Stable Minimal Plateau Surfaces

Thu, 20 Oct 2022 11:51:14 +0800

Main results

The classical Bernstein question is extended from smooth minimal surfaces to Plateau surfaces, whose local models may also contain $Y$- and $T$-type soap-film singularities.

Generalized Bernstein theorem

Stability and quadratic area growth force flatness

If a complete orientable minimal Plateau surface in $\mathbb R^3$ is stable and has at most quadratic area growth, then every component of its regular part lies in a plane.

No $T$-points

The surface is cylindrical over a planar network

When no $T$-type singularity occurs, the surface is, after a rigid motion, of the form $N\times\mathbb R$ for an embedded stationary network $N\subset\mathbb R^2$.

With $T$-points

Only two global configurations remain

If $T$-type singularities occur, there is either one such point and the surface is the standard $T$ cone, or there are exactly two points joined in the explicitly classified flat configuration.

Proof strategy

The proof transfers the stability inequality to the regular sheets while retaining the balancing information along the singular curves. Curvature estimates and logarithmic cutoffs, made possible by quadratic area growth, force the second fundamental form to vanish. Once each sheet is planar, the topology and Plateau angle conditions reduce the remaining work to a global classification of the straight $Y$-curves and isolated $T$-points.

Second Order Elliptic Operators on Triple Junction Surfaces

Mon, 03 Oct 2022 00:00:00 +0000

Slices can be found here.

General Theory of Partial Differential Equations on Triple Junction Surfaces

Sat, 02 Jul 2022 11:51:14 +0800

Main contributions

The thesis builds a general analytic framework for triple-junction hypersurfaces: singular spaces obtained by identifying the boundaries of several smooth sheets.

Function spaces

Analysis that remembers the junction geometry

Sobolev- and Hölder-type spaces are defined sheetwise with compatibility and balancing conditions along the common boundary, providing the correct domains for geometric variation problems.

Elliptic theory

Classical PDE tools extend to the singular setting

For second-order elliptic operators on triple-junction hypersurfaces, the thesis develops existence, regularity, Fredholm, and spectral results analogous to those on smooth Riemannian manifolds.

Applications

Geometry becomes accessible through the PDE framework

The theory is applied to Jacobi operators, Morse index questions for minimal triple junctions, and conformal and uniformization problems for singular surfaces.

Analytic viewpoint

The central principle is to separate local elliptic analysis on each smooth sheet from the finite-dimensional matching data along the junction. Trace conditions encode which functions arise from ambient variations, while the conormal balance supplies the natural transmission condition for second-order operators. Once regularity across that interface is established, variational, spectral, and geometric tools can be used much as in the smooth theory.

Curvature Estimates for Stable Minimal Surfaces with a Common Free Boundary

Thu, 19 May 2022 11:41:35 +0800

Main results

The paper extends curvature estimates and Bernstein-type rigidity from smooth minimal surfaces to weighted multiple-junction surfaces sharing a common free boundary.

Curvature estimate

An $L^p$ theory compatible with the junction

For stable minimal multiple-junction surfaces, the second fundamental forms satisfy an $L^{2p}$ estimate for $1

Compact junction

Quadratic growth forces every sheet to be flat

A complete, orientable, stable minimal multiple-junction surface in $\mathbb R^3$ with quadratic area growth, compact common boundary, and constant pairwise angles has flat sheets.

Straight junction

The same Bernstein conclusion holds along a line

For a triple junction meeting at $120^\circ$, the flatness conclusion also holds when the common boundary is a straight line, under the corresponding completeness, stability, and area-growth assumptions.

Proof strategy

Simons’ inequality is applied sheet by sheet and combined with the global stability inequality using test functions that satisfy the junction compatibility condition. The conormal boundary terms do not disappear automatically; the weighted balance law reorganizes them into a controllable expression. Iteration of the resulting $L^p$ estimate with large cutoffs and quadratic area growth forces the curvature to vanish.

Uniformization of surfaces with boundary and the application to the triple junction surfaces with negative Euler characteristic

Sat, 09 Oct 2021 11:51:14 +0800

Main results

The paper first develops a quantitative uniformization theorem for surfaces with boundary and then uses it to match the three sheets of a triple-junction surface.

Surfaces with boundary

Prescribed constant interior and boundary curvature

For a compact oriented surface with negative Euler characteristic and $(k,c)$ satisfying $k\leq0$ and $c<\sqrt{-k}$, there is a unique conformal metric with Gaussian curvature $k$ and constant boundary geodesic curvature $c$.

Quantitative control

Area and boundary length vary monotonically

The area $A(k,c)$ and boundary length $L(k,c)$ are continuous and satisfy sharp monotonicity and limiting laws, including the behavior of $cL(k,c)$ needed to balance junction curvature.

Triple junctions

A weak hyperbolic uniformization

If a triple-junction surface has connected junction and $\chi(M)\leq0$, its three sheets admit conformal hyperbolic metrics with a common boundary length whose geodesic curvatures sum to zero; the sheet metric is unique when its Euler characteristic is nonpositive.

Proof strategy

The prescribed-curvature problem is written as a semilinear elliptic boundary value problem. Maximum-principle estimates give existence, uniqueness, continuity, and monotonicity as $k$ and $c$ vary. For the junction problem, the boundary length is used as a common parameter: the three curvature-length functions are added, and their monotonicity and endpoint limits produce the unique balancing length.

Positive-curvature phenomenon

The following is a short video shows that why the uniqueness might fail when $K>0$.

In this video, we suppose the surface $\Sigma$ is the two copies of the domain enclosed by six arcs by identifying their corresponding edges marked with green color. So here $\Sigma$ is just a genus zero surface with three boundary components. We fix one boundary of $\Sigma$ having geodesic curvature $-1$ (marked as red) and the remaining two boundaries being geodesic (marked as black). Then we can let the area grow to see how the Gauss curvature changes after uniformization. The blue circle in this video represents the infinity circle if $K<0$ or the geodesic circle if $K>0$. From the video, we can see that $K$ will increase as area $A$ increases, then $K$ will reach its maximal point and after that $K$ will decrease to 0. So we will see that for some $K>0$, there are indeed two metrics with prescribed Gauss curvature and geodesic curvatures of boundaries.

Curvature Estimates for Stable Minimal Triple Junction Surfaces

Mon, 12 Jul 2021 00:00:00 +0000

Slices can be found here Part 1 Part 2.

Candidate \(P\)	Numerical screening
Rank-three tests: \(S=1-\nu_1^2-\nu_2^2-\nu_3^2,\quad t\in [0,2\pi]\)
\(P=(1-\nu_1)(1-\nu_2)\)	✓ Passed
\(P=\bigl[(1-\nu_1)(1-\nu_2)(1-\nu_1/\sqrt{2}-\nu_3/\sqrt{2})\bigr]^{2/3}\)	✕ Failed^*
\(P=\bigl[(1-\nu_1)(1-\nu_2)(1-\nu_1\cos t-\nu_3\sin t)\bigr]^{2/3}\)	✕ Failed^*
\(P=\bigl[(1-\nu_1)^2(1-\nu_2)\bigr]^{2/3}\)	✕ Failed^*
\(P=\bigl[(1-\nu_1)(1-\nu_2)(1-\nu_3)(1-\nu_1\cos t-\nu_3\sin t)\bigr]^{1/2}\)	✕ Failed^*
\(P=\bigl[(1-\nu_1)^2(1-\nu_2)(1-\nu_3)\bigr]^{1/2}\)	✕ Failed^*
\(P=\bigl[(1-\nu_1)^2(1-\nu_2)(1-\nu_2\cos t-\nu_3\sin t)\bigr]^{1/2}\)	✕ Failed^*
\(P=\left[\prod_{i=1}^2(1-\nu_i)(1-\nu_i\cos t-\nu_3\sin t)\right]^{1/2}\)	✕ Failed^*
\(P=\bigl[(1-\nu_1^2)(1-\nu_2^2)(1-\nu_3^2)\bigr]^{1/3}\)	✕ Failed^*
Additional \(n=6\) tests: \(S=1-\nu_1^2-\nu_2^2-\nu_3^2-\nu_4^2\)
\(P=\frac{1}{4}\bigl[(1-\nu_1)(1-\nu_2)(1-\nu_3)(1+\nu_3)\bigr]^2\)	✕ Failed
\(P=(1-\nu_1)(1-\nu_2)(1-\nu_3)\)	✕ Failed
\(P=(1-\nu_1)(1-\nu_2)\)	✕ Failed

Pole configuration	\(m\)	\(2/m\)	Boundary \(\min(\varrho/P)\)	Sampled \(\min\varrho\)
Coordinate triple	3	\(2/3\)	\(0.5006385\)	\(0.0999129\) ✓
Tetrahedral four	4	\(1/2\)	\(0.5000182\)	\(0.3344756\) ✓
Octahedral six	6	\(1/3\)	\(0.5000031\)	\(0.3157721\) ✓
Random four	4	\(1/2\)	\(0.5009658\)	\(0.0426013\) ✓
Random six	6	\(1/3\)	\(0.5000017\)	\(0.0427318\) ✓
Random eight	8	\(1/4\)	\(0.5010803\)	\(0.1085162\) ✓
Randomized two-pole (60 tries)	2	\(1\)	---	\(2.41\times10^{-1}\) ✓
Randomized three-pole (60 tries)	3	\(2/3\)	---	\(2.06\times10^{-1}\) ✓
Randomized four-pole (60 tries)	4	\(1/2\)	---	\(2.28\times10^{-1}\) ✓
Randomized five-pole (60 tries)	5	\(2/5\)	---	\(2.92\times10^{-1}\) ✓

Pole configuration	\(m\)	\(T\)	\(k\)	Sampled \(\min(W-s)\)
Coordinate triple	3	\(1.35\times10^{-4}\)	\(5.53\times10^{-5}\)	\(4.52\times10^{-5}\) ✓
Tetrahedral four	4	\(2.47\times10^{-6}\)	\(1.20\times10^{-8}\)	\(5.24\times10^{-6}\) ✓
Five-point cluster	5	\(3.90\times10^{-11}\)	\(4.10\times10^{-15}\)	\(1.17\times10^{-16}\) ✓
Five points with a close pair	5	\(1.77\times10^{-12}\)	\(6.40\times10^{-16}\)	\(6.39\times10^{-11}\) ✓
Octahedral six	6	\(8.28\times10^{-10}\)	\(1.20\times10^{-18}\)	\(2.78\times10^{-9}\) ✓

Candidate \(G=g^2\)	Numerical screening
\(G=1-\nu_1^2-\nu_2^2+k(\cos\vartheta-\nu_1)^2\)	✓ Passed
\(G=S+k\nu_1^2\nu_2^2\)	✕ Failed
\(G=S^2+k(\nu_1-\nu_2)^2(\nu_1+\nu_2-1)^2\)	✕ Failed
\(G=\bigl[S^2+k(\nu_1-\nu_2)^2(\nu_1+\nu_2-1)^2\bigr]^{1/2}\)	✕ Failed
\(G=\left[\prod_{\pm}\left(1-\frac{\sqrt{3}}{2}\nu_1\pm\frac{1}{2}\nu_2\right)(1+\nu_2)\right]^{2/3}\)	✓ Passed
\(G=\left[S^{3/2}+k\prod_{\pm}\left(1-\frac{\sqrt{3}}{2}\nu_1\pm\frac{1}{2}\nu_2\right)(1+\nu_2)\right]^{2/3}\)	✕ Failed
\(G=S+k(1-\nu_1^2)(1-\nu_2^2)\)	✕ Failed
\(G=\bigl[S^2+k(1-\nu_1^2)(1-\nu_2^2)\bigr]^{1/2}\)	✕ Failed
\(G=\bigl[(1-\nu_1)(1-\nu_2)(1-\nu_3)\bigr]^{2/3}\)	✓ Passed
\(G=\bigl[(1-\nu_1^2)(1-\nu_2^2)(1-\nu_3^2)(1-\nu_4^2)\bigr]^{1/4}\)	✕ Failed

Gaoming Wang

1. Purpose and perspective

Regularity and Compactness Results for Stable Minimal Hypersurfaces

Riemannian Manifolds

Simons’ Identity and Inequality

Stable Bernstein in Dimension Two

Properties of differential operators on complete Riemannian manifolds

The Operator \(\Delta-aK\) on Surfaces

Complete Stable Minimal Surfaces in 3-Manifolds

The conformal Laplacian

Varifolds

2. The capillary prototype

2.1 The boundary condition as a design equation

2.2 What was tried before the final family

2.3 The additive route to the final family

2.4 The design blueprint from the capillary problem

Classification of Stable Minimal Cones for n<= 6

Conformal descent on a stable minimal hypersurface

Higher-Dimensional Stable Bernstein Theorems

Conformal change of the metric on minimal hypersurfaces

Construction of the \(\mu\)-Bubble

Second variation of weighted area functional

Properties of the minimizer

Completion of the \(\mu\)-Bubble Construction

Diameter estimate and volume estimate under spectral Ricci curvature bound

Weighted geodesic proof of the diameter estimates.

Mazet’s proof in the ambient space \(\mathbb{R}^6\)

Key Steps in the Regularity Proofs

Sets of Finite Perimeter

Submanifolds

3. From the geometric inequality to a numerical criterion

Dimension Reduction and Regularity of Perimeter Minimizers

First and Second Variation Formulas

Geroch’s Conjecture by Dimension Reduction

Green-kernel proof of the stable Bernstein theorem in R^4

Regularity in the Immersed Setting

Sub-minimal Sets and Sub-calibrations

4. The first computational phase: explicit formulas

4.1 The first numerical candidates

4.2 Searching over the detector \(P\)

Interpreting the starred failures.

Bellettini’s Sheeting Theorem and Schoen–Simon Regularity

Proof of Bernstein Theorem

Properties of Minimal Submanifolds in R^N

The Positive Mass Theorem and the Reduction to Geroch’s Conjecture

5. The decisive small-parameter calculation

5.1 The small-\(k\) first-order expansion

5.2 The first-order criteria in ranks two and three

5.3 Rank-three scans and analytic confirmation

Monotonicity Formula

Sketch of Wickramasekera’s Regularity Theorem

The Horowitz–Myers Conjecture

From AdS energy to a Riemannian inequality

The Brendle–Hung mass inequality

The systolic boundary inequality

The AdS soliton and sharpness

How the Brendle–Hung proof works

6. The local repair and the final profile

6.1 The linear model near a zero

6.2 The logarithmic transition and the rank threshold

6.3 The final profile and its verification

Area-Minimizing Hypersurfaces and Calibration

7. Summary and outlook

Minimal Graphs

The Dirichlet Problem for the Minimal Surface Equation

Bernstein’s Theorem and Generalizations

Preface

References

References

Scalar-mean rigidity theorem and Llarull's theorem for nonspin manifolds

Main theorem

Llarull's theorem in all dimensions

Proof architecture

From the sphere to weighted scalar-mean comparison

Pass to a hypersurface and obtain a spectral inequality

Blow up the singular set while retaining the degree

Recover pointwise inequalities and reach dimension two

The stable Bernstein theorem in $\mathbb{R}^{7}$

Main theorem

Flatness in $\mathbb R^7$