The Penrose inequality with charge for 2-convex initial data sets

Abstract

We prove the Penrose inequality with charge under the 2-convexity condition recently introduced by Dong. More precisely, given a complete, connected and asymptotically flat Einstein-Maxwell initial data set $(M,g,k; E,B)$ satisfying the charged dominant energy and the 2-convexity conditions, with divergence-free electromagnetic vector fields $(E,B)$ and a connected outermost past apparent horizon $\Sigma$ that satisfies $|\Sigma| \geq 4\pi q^2$ - where $q$ is the total charge - we show that the following inequality for the ADM mass $m$ holds: $m\geq \sqrt{\frac{|\Sigma|}{16\pi}} + q^2 \sqrt{\frac{\pi}{|\Sigma|}}$, with equality if and only if $k \equiv 0$ and $(M,g;E,B)$ is isometric to a canonical slice of sub-extremal Reissner-Nordstr\"om spacetime. Building on Dong's proof of the uncharged case, we use his $\mathbf{P}$-inverse mean curvature flow and its weak formulation, which only depends on $(g,\mathbf{P})$ and hence applies to the charged setting unchanged. The novelty of our work is the modification of the monotonicity formula to account for the additional charge term. For time-symmetric data ($k \equiv 0$), the flow reduces to the classical inverse mean curvature flow and our monotonicity formula to Jang's monotonicity of the charged Hawking mass, recovering the charged Riemannian Penrose inequality.