Geometric inequalities for critical metrics of the volume functional

TL;DR

This paper studies the geometric properties of critical metrics related to the volume functional on compact manifolds, providing sharp estimates for boundary mean curvature and area, including localized versions.

Contribution

It introduces new sharp estimates for boundary mean curvature and area of critical metrics of the volume functional on compact manifolds, including localized results.

Findings

Sharp estimates for boundary mean curvature

Sharp estimates for boundary area

Localized boundary estimates

Abstract

In this article, we investigate the geometry of critical metrics of the volume functional on an $n$-dimensional compact manifold with (possibly disconnected) boundary. We establish sharp estimates to the mean curvature and area of the boundary components of critical metrics of the volume functional on a compact manifold. In addition, localized version estimates to the mean curvature and area of the boundary of critical metrics are also obtained.