Mass, Conformal Capacity, and the Volumetric Penrose Inequality

Let $\mathrm{\Omega }$ be a smooth, bounded subset of ${\mathbb{R}}^3$ diffeomorphic to a ball. Consider $M={\mathbb{R}}^3\setminus \mathrm{\Omega }$ 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 $\mathrm{\Sigma }=\mathrm{\partial }M$ 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 $\mathrm{\Omega }$. As a corollary, we deduce a sharp lower bound for the ADM mass of $M$ in terms of the Euclidean volume of $\mathrm{\Omega }$. 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.