Stability | Gaoming Wang

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 $A\equiv0$.

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

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.

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.

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.

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.

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.