A volumetric Penrose inequality for conformally flat manifolds

Abstract

We consider asymptotically flat Riemannian manifolds with nonnegative scalar curvature that are conformal to ${\mathbb{R}}^n\setminus \Omega ,n\ge 3$, and so that their boundary is a minimal hypersurface. (Here, $\Omega \subset {\mathbb{R}}^n$ 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.

7 pages

References in corpus (1)

• On The Capacity of Surfaces in Manifolds with Nonnegative Scalar Curvature