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

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 $3\le \dim (M)\le 7$, $(M,g)$ has non-negative scalar curvature, and the boundary $\partial M$ is mean-convex, we obtain the Riemannian Penrose-type inequality $$m\ge {\left(\frac{1}{2}\right)}^{\frac{n}{n-1}}{\left(\frac{|\Sigma |}{{\omega }_{n-1}}\right)}^{\frac{n-2}{n-1}}$$ 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.

The final version has appeared in Comm. Math. Phys

References in corpus (4)

• Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature
• Initial data rigidity results
• Spacetime positive mass theorems for initial data sets with noncompact boundary
• The Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary