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

Abstract

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 will give the complete classification for an $n$-dimensional, $n=3$ or $4,$ weakly Einstein critical metric of the volume functional with nonnegative scalar curvature. 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.