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

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 derivative 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 >= 5), we will establish a similar result for weakly Einstein critical metric under a suitable constraint on the Weyl tensor. (C) Elsevier Inc. All rights reserved.