Doubling of Asymptotically Flat Half-spaces and the Riemannian Penrose Inequality

Abstract

AbstractBuilding on previous works of Bray, of Miao, and of Almaraz, Barbosa, and de Lima, we develop a doubling procedure for asymptotically flat half-spaces (M, g) with horizon boundary $$\Sigma \subset M$$ Σ ⊂ M and mass $$m\in {\mathbb {R}}$$ m ∈ R . If $$3\le \dim (M)\le 7$$ 3 ≤ dim ( M ) ≤ 7 , (M, g) has non-negative scalar curvature, and the boundary $$\partial M$$ ∂ M is mean-convex, we obtain the Riemannian Penrose-type inequality $$\begin{aligned} m\ge \left( \frac{1}{2}\right) ^{\frac{n}{n-1}}\,\left( \frac{|\Sigma |}{\omega _{n-1}}\right) ^{\frac{n-2}{n-1}} \end{aligned}$$ m ≥ 1 2 n n - 1 | Σ | ω n - 1 n - 2 n - 1 as a corollary. Moreover, in the case where $$\partial M$$ ∂ 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 $$\dim (M)=3$$ dim ( M ) = 3 and $$\Sigma $$ Σ is a connected free boundary hypersurface.