On the volume functional of compact manifolds with harmonic Weyl tensor

Abstract

AbstractThe main aim of this article is to give the complete classification of critical metrics of the volume functional on a compact manifold M with boundary and under the harmonic Weyl tensor condition. In particular, we prove that a critical metric with a harmonic Weyl tensor on a simply connected compact manifold with the boundary isometric to a standard sphere must be isometric to a geodesic ball in a simply connected space form , , and . To this end, we first conclude the classification of such critical metrics under the Bach‐flat assumption and then we prove that both geometric conditions are equivalent in this situation.