Einstein and conformally flat critical metrics of the volume functional

Abstract

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