Einstein and conformally flat critical metrics of the volume functional

Abstract

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