A Heintze-Karcher type inequality in hyperbolic space

TL;DR

This paper establishes a new geometric inequality in hyperbolic space and applies it to prove uniqueness and classification results for hypersurfaces with constant curvature properties.

Contribution

It introduces a Heintze-Karcher type inequality in hyperbolic space and derives new Alexandrov and uniqueness theorems for specific hypersurfaces.

Findings

Proved a new Heintze-Karcher inequality in hyperbolic space.

Established an Alexandrov type theorem for constant shifted mean curvature hypersurfaces.

Obtained a uniqueness result for h-convex hypersurfaces with certain curvature conditions.

Abstract

In this paper, we prove a new Heintze-Karcher type inequality for shifted mean convex hypersurfaces in hyperbolic space. As applications, we prove an Alexandrov type theorem for closed embedded hypersurfaces with constant shifted $k$th mean curvature in hyperbolic space. Furthermore, a uniqueness result for $h$-convex hypersurfaces satisfying certain curvature equations is obtained.