Doubling of asymptotically flat half-spaces and the Riemannian Penrose inequality

TL;DR

This paper develops a doubling procedure for asymptotically flat half-spaces with horizon boundary, establishing a Riemannian Penrose inequality in higher dimensions and characterizing equality cases as Schwarzschild half-spaces.

Contribution

It extends the Riemannian Penrose inequality to higher dimensions for asymptotically flat half-spaces with boundary, including new perturbation techniques for scalar curvature.

Findings

Proves the Penrose inequality for dimensions 3 to 7.

Characterizes equality cases as Schwarzschild half-spaces.

Introduces local perturbations increasing scalar curvature.

Abstract

Building on previous works of H. L. Bray, of P. Miao, and of S. Almaraz, E. Barbosa, and L. L. de Lima, we develop a doubling procedure for asymptotically flat half-spaces $(M,g)$ with horizon boundary $\Sigma\subset M$ and mass $m\in\mathbb{R}$. If $3\leq \dim(M)\leq 7$, $(M,g)$ has non-negative scalar curvature, and the boundary $\partial M$ is mean-convex, we obtain the Riemannian Penrose-type inequality $$ m\geq\left(\frac{1}{2}\right)^{\frac{n}{n-1}}\,\left(\frac{|\Sigma|}{\omega_{n-1}}\right)^{\frac{n-2}{n-1}} $$ as a corollary. Moreover, in the case where $\partial M$ is not totally geodesic, we show how to construct local perturbations of $(M,g)$ that increase the scalar curvature. As a consequence, we show that equality holds in the above inequality if and only if the exterior region of $(M,g)$ is isometric to a Schwarzschild half-space. Previously, these results were only known in the case where $\dim(M)=3$ and $\Sigma$ is a connected free boundary hypersurface.