A volumetric Penrose inequality for conformally flat manifolds

TL;DR

This paper proves a lower bound on the ADM mass of certain conformally flat, asymptotically flat manifolds with minimal boundary, relating it to the volume of a bounded domain, advancing a conjecture by Bray and Iga.

Contribution

It establishes a volumetric Penrose inequality for conformally flat manifolds with minimal boundary, without requiring the boundary to be outermost.

Findings

The ADM mass is bounded below by a function of the volume of the bounded domain.

The result applies to manifolds conformal to Euclidean space minus a bounded domain.

It provides partial proof of a conjecture by Bray and Iga.

Abstract

We consider asymptotically flat Riemannian manifolds with nonnegative scalar curvature that are conformal to $\R^{n}\setminus \Omega, n\ge 3$, and so that their boundary is a minimal hypersurface. (Here, $\Omega\subset \R^{n}$ is open bounded with smooth mean-convex boundary.) We prove that the ADM mass of any such manifold is bounded below by $(V/\beta_{n})^{(n-2)/n}$, where $V$ is the Euclidean volume of $\Omega$ and $\beta_{n}$ 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.