Mass and volume of four-dimensional Einstein metrics

TL;DR

This paper investigates the mass and volume of four-dimensional Einstein metrics, deriving integral formulas for ADM mass and establishing inequalities and characterizations for specific Einstein manifolds.

Contribution

It introduces a new integral expression for the ADM mass of conformally related metrics and proves inequalities linking mass and volume, characterizing special Einstein metrics.

Findings

Derived an integral formula for ADM mass in terms of the conformal Green's function.

Established a lower bound for mass based on volume of the Einstein manifold.

Proved mass gap theorems characterizing the round sphere and Fubini-Study 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 $\mathbb{CP}^2$.