Some rigidity results on compact hypersurfaces with capillary boundary in Hyperbolic space

TL;DR

This paper establishes rigidity results and an Alexandrov type theorem for capillary hypersurfaces in hyperbolic space, extending geometric inequalities and characterizing equality cases for such hypersurfaces.

Contribution

It proves a Heintze-Karcher inequality for capillary hypersurfaces in hyperbolic space and characterizes the equality case, leading to a new Alexandrov type theorem.

Findings

Heintze-Karcher inequality for capillary hypersurfaces in hyperbolic space

Characterization of equality cases as totally umbilical hypersurfaces

Rigidity results for hypersurfaces supported on totally geodesic planes

Abstract

In this paper, we prove a Heintze-Karcher type inequality for capillary hypersurfaces supported on various hypersurfaces in the hyperbolic space. The equality case only occurs on capillary totally umbilical hypersurfaces. Then we apply this result to prove the Alexandrov type theorem for embedded capillary hypersurfaces in the hyperbolic space. In addition, we prove some other rigidity results for capillary hypersurfaces supported on totally geodesic plane in $\mathbb B^{n+1}_+$.