  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
$\mathcal{M}^R_\gamma$.
