Regularity | Gaoming Wang

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.

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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.

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.