Remark on a geometric inequality for closed hypersurfaces in weighted manifolds

TL;DR

This paper establishes sharp geometric inequalities for the boundary of bounded sets in noncompact weighted manifolds with nonnegative Bakry-Émery Ricci curvature, generalizing classical inequalities to the weighted setting.

Contribution

It introduces sharp boundary inequalities in weighted manifolds with nonnegative Ricci curvature, extending classical results and characterizing rigidity cases.

Findings

Inequalities are sharp for both finite and infinite N.

Rigidity occurs when the complement of the set is a twisted product.

Generalizes Willmore-type inequality to weighted manifolds.

Abstract

In this paper we consider noncompact smooth metric measure spaces $(M, g,e^{-f}dvol_{g})$ of nonnegative Bakry-\'Emery Ricci curvature, i.e. $Ric + D^{2}f - \frac{1}{N}df \otimes df \geq 0$, for $0< N \leq \infty$, in order to obtain geometric inequalities for the boundary of a given open and bounded set $\Omega\subset M$, with regular boundary $\partial \Omega$. Our inequalities are sharp for both the cases $N< \infty$ and $N= \infty$, provided that the underlying ambient space has large weighted volume growth. The rigidity obtained for the $N=\infty$ case holds true precisely when $M \setminus \Omega$ is isometric to a twisted product metric and, as such, is a generalization of the Willmore-type inequality for nonnegative Ricci curvature from Agostiniani, Fagagnolo and Mazzieri to the context of weighted manifolds.