The Spacetime Penrose Inequality for Cohomogeneity One Initial Data

Abstract

We prove the spacetime Penrose inequality for asymptotically flat $2(n+1)$-dimensional initial data sets for the Einstein equations, which are invariant under a cohomogeneity one action of $\mathrm{SU}(n+1)$. Analogous results are obtained for asymptotically hyperbolic initial data that arise as spatial hypersurfaces in asymptotically Anti de-Sitter spacetimes. More precisely, it is shown that with the dominant energy condition, the total mass is bounded below by an explicit function of the outermost apparent horizon area. Furthermore, the inequality is saturated if and only if the initial data isometrically embed into a Schwarzschild(-AdS) spacetime. This generalizes the only previously known case of the conjectured spacetime Penrose inequality, established under the assumption of spherical symmetry. Additionally, in the time symmetric case, we observe that the inequality holds for $4(n+1)$-dimensional and 16-dimensional initial data invariant under cohomogeneity one actions of $\mathrm{Sp}(n+1)$ and $\mathrm{Spin}(9)$, respectively, thus treating the inequality for all cohomogeneity one actions in this regime.