Regularity of stable capillary minimal hypersurfaces

TL;DR

This paper develops a regularity and compactness theory for stable capillary minimal hypersurfaces in half-space, introducing new integral curvature estimates and boundary sheeting techniques, leading to generalized Bernstein theorems.

Contribution

It introduces a novel integral curvature estimate and boundary sheeting theorem for stable capillary minimal hypersurfaces, advancing regularity and classification results.

Findings

Established a boundary sheeting theorem for stable capillary minimal hypersurfaces.

Derived a generalized Bernstein theorem for embedded complete stable capillary minimal hypersurfaces.

Developed an integral curvature estimate that eliminates boundary terms in stability inequalities.

Abstract

We develop a regularity and compactness theory for stable capillary minimal hypersurfaces in the half-space $\mathbb{H}^{n+1}$ with contact angle $\theta \in (0,\pi)$ and dimension $n \geq 2$. As a consequence, we obtain the generalized Bernstein theorem for embedded complete stable capillary minimal hypersurfaces in $\mathbb{H}^{n+1}$ with Euclidean area growth. The key innovation is an integral curvature estimate: by carefully selecting an appropriate tilt excess function, we are able to eliminate the boundary terms arising in the stability inequality. Building on this, we establish a boundary sheeting theorem by refining the arguments in [SS81]. These results, combined with a refined classification of stable capillary minimal cones, lead to the main regularity and compactness theorems.