Weighted isoperimetric inequalities in warped product manifolds

TL;DR

This paper establishes new isoperimetric inequalities in warped product manifolds and connects them to curvature integrals, eigenvalue estimates, and geometric inequalities in space forms.

Contribution

It introduces novel isoperimetric inequalities in warped product manifolds and relates them to higher order mean-curvature integrals and eigenvalue bounds.

Findings

Derived sharp isoperimetric inequalities in warped product manifolds

Connected geometric inequalities to higher order mean-curvature integrals

Obtained sharp eigenvalue estimates in space forms

Abstract

We prove some isoperimetric type inequalities in warped product manifolds, or more generally, multiply warped product manifolds. We then relate them to inequalities involving the higher order mean-curvature integrals. We also apply our results to obtain sharp eigenvalue estimates and some sharp geometric inequalities in space forms.