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.

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.

Capillary Surface | Gaoming Wang

Regularity of stable capillary minimal hypersurfaces

Main results

Angle-dependent sharp dimension bounds

Flatness for every contact angle

Flatness for $2\leq n\leq4$

Proof architecture

Scalar curvature rigidity of domains in a 3-dimensional warped product

Main results

Metric, scalar, and boundary comparison determine the model

Smooth, conical, and disk-type ends are treated uniformly

Antipodal cone points are allowed

Proof strategy