Stable capillary hypersurfaces supported on a horosphere in the hyperbolic space

TL;DR

This paper investigates the stability of free boundary and capillary hypersurfaces in hyperbolic space, establishing that only umbilical hypersurfaces and totally geodesic hyperplanes are stable under certain boundary conditions.

Contribution

It proves the uniqueness of stable immersed capillary hypersurfaces and hyperplanes supported on a horosphere in hyperbolic space.

Findings

Umbilical hypersurfaces are the only stable immersed capillary hypersurfaces with boundary on a horosphere.

A totally geodesic hyperplane is the only stable immersed type-II hypersurface with boundary on a horosphere.

The methods used can classify stability of hypersurfaces in hyperbolic space based on boundary conditions.

Abstract

In this paper, we study a stability problem of free boundary hypersurfaces, and also capillary ones whose boundary supported on a horosphere in hyperbolic space. We prove that umbilical hypersurfaces are only stable immersed capillary hypersurfaces whose boundary supported on a horosphere. Using the same method, we show that a totally geodesic hyperplane is only stable immersed type-II hypersurface whose boundary supported on a horosphere.