Mass and volume of four-dimensional Einstein metrics

Abstract

Let $(M^4,\bar{g})$ be an Einstein manifold, where $M^4$ is a smooth, closed, oriented four-manifold $M^4$ and $\bar{g}$ has positive Einstein constant. Given a point $0 \in M^4$, let $G$ denote the (positive) Green's function $G$ of the conformal laplacian $L_{\bar{g}}$; then $g = G^2 \bar{g}$ is a complete, scalar-flat, asymptotically flat metric on $\widehat{M} = M \setminus \{ 0 \}$. We first show that the ADM mass of $g$ can be expressed as an integral over $\widehat{M}$, then use this identity to prove a lower bound for the mass of $g$ in terms of the volume of $\bar{g}$. As corollaries, we prove a 'mass times volume' inequality, plus various mass gap theorems characterizing the round metric on $S^4$ and the Fubini-Study metric on $\mathbb{CP}^2$.