Stability of hypersurfaces with constant mean curvature trapped between two parallel hyperplanes

TL;DR

This paper investigates the stability of constant mean curvature hypersurfaces, including liquid bridges, between two parallel hyperplanes across all dimensions, using second variation analysis and numerical methods.

Contribution

It provides the first comprehensive stability analysis of non-uniform liquid bridges in all dimensions and parameter regimes.

Findings

Determined stability conditions for hypersurfaces with boundaries on hyperplanes.

Extended stability results of liquid bridges to arbitrary dimensions.

Used numerical computations to support analytical findings.

Abstract

Static equilibrium configurations of continua supported by surface tension are given by constant mean curvature (CMC) surfaces which are critical points of a variational problem to extremize the area while keeping the volume fixed. CMC surfaces are used as mathematical models of a variety of continua, such as tiny liquid drops, stars, and nuclei, to play important roles in both mathematics and physics. Therefore, the geometry of CMC surfaces and their properties such as stability are of special importance in differential geometry and in a variety of physical sciences. In this paper we examine the stability of CMC hypersurfaces in arbitrary dimensions, possibly having boundaries on two parallel hyperplanes, by investigating the second variation of the area. We determine the stability of non-uniform liquid bridges or unduloids for the first time in all dimensions and all parameter (the ratio of the neck radius to bulge radius) regimes. The analysis is assisted by numerical computations.