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 ∂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 established a similar result for weakly Einstein critical metric under a suitable constraint on the Weyl tensor.2010 Mathematics Subject Classification. Primary 53C25; Secondary 53C20, 53C21, 53C24.

Paracontact Metric κ , μ -Manifold Satisfying the Miao-Tam Equation

“…And then several generalizations of this rigidity result were found by different authors, replacing the Einstein assumption by a weaker condition such as harmonic Weyl tensor [3], parallel Ricci tensor [4], or cyclic parallel Ricci tensor [5]. For Some other generalizations or rigidity results, we can refer to [6][7][8][9][10], etc.…”