Stable capillary hypersurfaces in a half-space or a slab

TL;DR

This paper proves that stable capillary hypersurfaces in a half-space or slab are rotationally symmetric under various conditions, revealing their geometric structure and classification in Euclidean space.

Contribution

It establishes symmetry results for stable capillary hypersurfaces in half-spaces and slabs, extending understanding of their geometric properties and classifications.

Findings

Hypersurfaces are rotationally symmetric in specified cases.

In case (2), non-cylindrical hypersurfaces are graphical over boundary domains.

In case (3), hypersurfaces are spherical caps.

Abstract

We study stable immersed capillary hypersurfaces in a domain $\mathcal B$ which is either a half-space or a slab in the Euclidean space $\Bbb R^{n+1}.$ We prove that such a hypersurface $\Sigma$ is rotationally symmetric in the following cases: (1) $n=2$, $\mathcal B$ is a slab and $\Sigma$ has genus zero, (2) $n\geq 2$, $\mathcal B$ is a slab, the angle of contact is $\pi/2$ and each component of $\partial\Sigma$ is embedded, (3) $n\geq 2,$ $\mathcal B$ is a half-space, the angle of contact is $<\pi/2$ and each component of $\partial\Sigma$ is embedded. Moreover, in case (2), if not a right circular cylinder, the hypersurface has to be graphical over a domain in $\partial\mathcal B.$ In case (3), the hypersurface is a spherical cap.