Mass, Conformal Capacity, and the Volumetric Penrose Inequality

Abstract

Let $\Omega$ be a smooth, bounded subset of ${\mathbb{R}}^3$ diffeomorphic to a ball. Consider $M={\mathbb{R}}^3\setminus \Omega$ equipped with an asymptotically flat metric $g=f^4{g}_{\text{euc}}$, where $f\to 1$ at infinity. Assume that $g$ has non-negative scalar curvature and that $\Sigma =\partial M$ is a minimal 2-sphere in the $g$ metric. We prove a sharp inequality relating the ADM mass of $M$ with the conformal capacity of $\Omega$. As a corollary, we deduce a sharp lower bound for the ADM mass of $M$ in terms of the Euclidean volume of $\Omega$. We also prove a stability type result for this ``volumetric Penrose inequality.'' The proofs are based on a monotonicity formula holding along the level sets of a 3-harmonic function.

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