Integral and boundary estimates for critical metrics of the volume functional

TL;DR

This paper derives new sharp integral and boundary estimates for critical metrics of the volume functional on compact manifolds with boundary, using generalized Reilly's formula to advance geometric analysis in this context.

Contribution

It introduces novel integral and boundary estimates for critical volume metrics on compact manifolds with boundary, expanding the understanding of their geometric properties.

Findings

Derived sharp integral estimates for critical volume metrics.

Established new boundary estimates for these metrics.

Applied generalized Reilly's formula to achieve these results.

Abstract

In this article, we investigate the geometry of critical metrics of the volume functional on compact manifolds with boundary. We use the generalized Reilly's formula to derive new sharp integral estimates for critical metrics of the volume functional on $n$-dimensional compact manifolds with boundary. As application, we establish new boundary estimates for such manifolds.