Stable $(r+1)$-th capillary hypersurfaces

Abstract

In this paper, we propose a new definition of stable $(r+1)$-th capillary hypersurfaces from variational perspective for any $1\le r\le n-1$. More precisely, we define stable $(r+1)$-th capillary hypersurfaces to be smooth local minimizers of a new energy functional under volume-preserving and contact angle-preserving variations. Using this new concept of stable $(r+1)$-th capillary hypersurfaces, we generalize the stability results of Souam (2023) in a Euclidean half-space, and Guo, Wang and Xia (2022) in a horoball in hyperbolic space for capillary hypersurfaces to the $(r+1)$-th capillary hypersurface case.