On the volume functional of compact manifolds with harmonic Weyl tensor

The 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 $\partial M$ 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 ${\mathbb{S}}^{n-1}$ must be isometric to a geodesic ball in a simply connected space form ${\mathbb{R}}^n$, ${\mathbb{H}}^n$, and ${\mathbb{S}}^n$. 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.