Einstein and conformally flat critical metrics of the volume functional

Abstract:

Let $R$ be a constant. Let $\mathcal {M}^R_\gamma$ be the space of smooth metrics $g$ on a given compact manifold $\Omega ^n$ ($n\ge 3$) with smooth boundary $\Sigma$ such that $g$ has constant scalar curvature $R$ and $g|_{\Sigma }$ is a fixed metric $\gamma$ on $\Sigma$. Let $V(g)$ be the volume of $g\in \mathcal {M}^R_\gamma$. In this work, we classify all Einstein or conformally flat metrics which are critical points of $V( \cdot )$ in $\mathcal {M}^R_\gamma$.