Stable capillary hypersurfaces in a wedge

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 $\partial\Sigma$.