Bach-flat critical metrics of the volume functional on 4-dimensional manifolds with boundary

Abstract

The purpose of this article is to investigate Bach-flat critical metrics of the volume functional on a compact manifold $M$ with boundary $\partial M.$ Here, we prove that a Bach-flat critical metric of the volume functional on a simply connected 4-dimensional manifold with boundary isometric to a standard sphere must be isometric to a geodesic ball in a simply connected space form $\Bbb{R}^{4},$ $\Bbb{H}^{4}$ or $\Bbb{S}^{4}$. Moreover, we show that in dimension three the result even is true replacing the Bach-flat condition by the weaker assumption that $M$ has divergence-free Bach tensor.