Einstein and conformally flat critical metrics of the volume functional

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