Regularity of stable capillary minimal hypersurfaces

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 $n\ge 2$. One key analytic ingredient is a capillary differential Schoen inequality, which allows us to establish a boundary sheeting theorem in the spirit of Bellettini. The other ingredient is a refined classification of stable capillary minimal cones, and we show that for any contact angle $\theta \in (0,\pi )$, the stable capillary minimal hypercone in ${\mathbb{H}}^5$ with an isolated singularity must be flat. As a consequence, for any $n\le 4$ and $\theta \in (0,\pi )$, we obtain the Bernstein theorem for embedded complete stable capillary minimal hypersurfaces in ${\mathbb{H}}^{n+1}$ with Euclidean area growth.

53 pages, 3 figures. This revision classifies stable capillary minimal cones in H^5 for all contact angles, thereby raising the general-angle regularity threshold by one dimension. It also provides a shorter proof of the boundary sheeting theorem, based on a capillary two-level Caccioppoli inequality and the De Giorgi iteration used by Bellettini