Alexandrov-Fenchel inequalities for capillary hypersurfaces in hyperbolic space

TL;DR

This paper establishes Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in hyperbolic space by introducing quermassintegrals and employing a novel inverse curvature flow, extending previous results to new boundary conditions.

Contribution

It introduces quermassintegrals for hypersurfaces with capillary boundaries and proves new inequalities using a locally constrained inverse curvature flow, generalizing prior theorems.

Findings

Established Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in hyperbolic space.

Developed a new locally constrained inverse curvature flow method.

Extended previous results to hypersurfaces with capillary boundary conditions.

Abstract

In this article, we first introduce the quermassintegrals for compact hypersurfaces with capillary boundaries in hyperbolic space from a variational viewpoint, and then we solve an isoperimetric type problem in hyperbolic space. By constructing a new locally constrained inverse curvature flow, we obtain the Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in hyperbolic space. This generalizes a theorem of Brendle-Guan-Li \cite{BGL} for convex closed hypersurfaces in hyperbolic space.