Weakly Einstein critical metrics of the volume functional on compact manifolds with boundary

H. Baltazar, A. Da Silva, F. Oliveira

Published 2018-04-27Version 1

The goal of this paper is to study weakly Einstein critical metrics of the volume functional on a compact manifold $M$ with smooth boundary $\partial M$. Here, we prove that an $n$-dimensional, $n=3$ or $4,$ weakly Einstein critical metric of the volume functional with nonnegative scalar curvature must be isometric to a geodesic ball in a simply connected space form $\mathbb{R}^{n}$ or $\mathbb{S}^{n}$. Moreover, in the higher dimensional case ($n\geq5$), we will established a similar result for weakly Einstein critical metric under a suitable constraint on the Weyl tensor.