A volumetric Penrose inequality for conformally flat manifolds

Abstract

We consider asymptotically flat Riemannian manifolds with nonnegative scalar curvature that are conformal to $\R^{n}\setminus Ω, n\ge 3$, and so that their boundary is a minimal hypersurface. (Here, $Ω\subset \R^{n}$ is open bounded with smooth mean-convex boundary.) We prove that the ADM mass of any such manifold is bounded below by $(V/β_{n})^{(n-2)/n}$, where $V$ is the Euclidean volume of $Ω$ and $β_{n}$ is the volume of the Euclidean unit $n$-ball. This gives a partial proof to a conjecture of Bray and Iga \cite{brayiga}. Surprisingly, we do not require the boundary to be outermost.