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