Stable capillary hypersurfaces in a wedge

TL;DR

This paper proves that stable capillary hypersurfaces in a wedge or half-space with certain boundary conditions are spherical segments, extending classical results to higher dimensions and specific boundary configurations.

Contribution

It establishes that under stability and boundary convexity or embedding conditions, such hypersurfaces must be spherical, generalizing known results to higher dimensions and wedge geometries.

Findings

Stable hypersurfaces are spherical under given conditions.

Results apply to both wedge and half-space geometries.

Conditions include boundary convexity or embedding and constant contact angles.

Abstract

Let $\Sigma$ be a compact immersed stable capillary hypersurface in a wedge bounded by two hyperplanes in $\mathbb R^{n+1}$. Suppose that $\Sigma$ meets those two hyperplanes in constant contact angles and is disjoint from the edge of the wedge. It is proved that if $\partial \Sigma$ is embedded for $n=2$, or if $\partial\Sigma$ is convex for $n\geq3$, then $\Sigma$ is part of the sphere. And the same is true for $\Sigma$ in the half-space of $\mathbb R^{n+1}$ with connected boundary $\partial\Sigma$.