A class of weighted isoperimetric inequalities in hyperbolic space

TL;DR

This paper establishes new weighted isoperimetric inequalities in hyperbolic space and related manifolds, extending previous results by removing convexity assumptions and applying to star-shaped domains.

Contribution

It introduces a novel class of weighted isoperimetric inequalities in hyperbolic space, removing prior convexity restrictions and applying to star-shaped domains in warped product manifolds.

Findings

Proved weighted isoperimetric inequalities in hyperbolic space.

Removed the horo-convex assumption for certain domains.

Extended inequalities to star-shaped hypersurfaces in anti-de Sitter-Schwarzschild manifold.

Abstract

In this paper, we prove a class of weighted isoperimetric inequalities for bounded domains in hyperbolic space by using the isoperimetric inequality with log-convex density in Euclidean space. As a consequence, we remove the horo-convex assumption of domains in a weighted isoperimetric inequality proved by Scheuer-Xia. Furthermore, we prove weighted isoperimetric inequalities for star-shaped domains in warped product manifolds. Particularly, we obtain a weighted isoperimetric inequality for star-shaped hypersurfaces lying outside a certain radial coordinate slice in the anti-de Sitter-Schwarzschild manifold.