On stable capillary hypersurfaces with planar boundaries

TL;DR

This paper investigates the stability and shape of capillary hypersurfaces with planar boundaries, proving they are spherical caps under certain conditions, especially in domains bounded by hyperplanes with specific contact angles.

Contribution

It establishes conditions under which stable capillary hypersurfaces in hyperplane-bounded domains are spherical caps, extending understanding of their geometric properties.

Findings

In a half-space, stable hypersurfaces are spherical caps.

In domains bounded by multiple hyperplanes, hypersurfaces are spherical caps if contact angles are near right angles.

Provides conditions ensuring hypersurfaces are spherical based on boundary geometry and contact angles.

Abstract

We study stable immersed capillary hypersurfaces $\Sigma$ in domains B of R n+1 bounded by hyperplanes. When B is a half-space, we show $\Sigma$ is a spherical cap. When B is a domain bounded by k hyperplanes P 1 ,. .. , P k , 2 $\le$ k $\le$ n + 1, having independent normals, and $\Sigma$ has contact angle $\theta$ i with P i and does not touch the vertices of B, we prove there exists $\delta$ > 0, depending only on P 1 ,. .. , P k , so that if $\theta$ i $\in$ ($\pi$ 2 -- $\delta$, $\pi$ 2 + $\delta$) for each i, then $\Sigma$ has to be a piece of a sphere.