Integral and boundary estimates for critical metrics of the volume functional

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.

Geometry of static perfect fluid space-time

“…Subsequently, Xia [49] used the generalized Reilly's formula to establish a Minkowski type inequality for weighted mixed volumes in non-Euclidean space forms. Very recently, Diógenes et al [18] used the generalized Reilly's formula by Qiu and Xia to obtain sharp integral estimates for critical metrics of the volume functional that were used to obtain a sharp boundary estimate for such metrics.…”