Willmore-type inequalities for closed hypersurfaces in weighted manifolds

TL;DR

This paper establishes new Willmore-type inequalities for closed hypersurfaces in weighted manifolds with nonnegative Bakry-Émery Ricci curvature, including sharp inequalities in steady and shrinking gradient Ricci solitons, with characterizations of equality cases.

Contribution

It introduces weighted versions of Willmore inequalities for hypersurfaces in specific Ricci soliton backgrounds, extending previous unweighted results.

Findings

Sharp Willmore inequalities in steady gradient Ricci solitons

Sharp Willmore inequalities in shrinking gradient Ricci solitons

Characterization of equality cases for these inequalities

Abstract

In this paper, we prove some Willmore-type inequalities for closed hypersurfaces in weighted manifolds with nonnegative Bakry-\'Emery Ricci curvature. In particular, we give a sharp Willmore type inequality in steady gradient Ricci solitons. We also prove a sharp Willmore-like inequality in shrinking gradient Ricci solitons. Moreover, we characterize the equality cases of Willmore-type inequalities. These results can be regarded as weighted versions of Agostiniani-Fogagnolo-Mazzieri's Willmore-type inequality. As applications, we derive some sharp isoperimetric type inequalities in weighted manifolds under the existence assumption of a critical set of weighted isoperimetric functional.