Quasi-Local Penrose Inequalities with Electric Charge

Abstract

The Riemannian Penrose inequality is a remarkable geometric inequality between the ADM mass of an asymptotically flat manifold with nonnegative scalar curvature and the area of its outermost minimal surface. A version of the Riemannian Penrose inequality has also been established for the Einstein–Maxwell equations where the lower bound on mass depends also on electric charge, a charged Riemannian Penrose inequality. Here, we establish some quasi-local charged Penrose inequalities for surfaces isometric to closed surfaces in a suitable Reissner–Nordström manifold, which serves as a reference manifold for the quasi-local mass. In the case where the reference manifold has zero mass and nonzero electric charge, the lower bound on quasi-local mass is exactly the lower bound on the ADM mass given by the charged Penrose inequality.

Fill-ins with scalar curvature lower bounds and applications to positive mass theorems

“…Therefore the appropriate Bartnik data for including electric charge is the triple ( , g, H , φ) (see, for example, Sect. 3 of [6]).…”

“…Proof We apply Theorem 5.1 to construct a fill-in of the Bartnik data ( , g, H o , φ), where H o = max (H ). We then obtain a (charged) manifold with corner by attaching the fill-in to M and can apply the charged Riemannian Penrose inequality for manifolds with corners established in [6]. Note that the results in [6] are stated only in dimension 3 only, however it is clear from the proof that the charged Riemannian Penrose inequality with corners holds in dimension up to 7 (see remark 5.4 below).…”

“…Note that the results in [6] are stated only in dimension 3 only, however it is clear from the proof that the charged Riemannian Penrose inequality with corners holds in dimension up to 7 (see remark 5.4 below). Specifically, from Theorem 1.3 of [6] we have…”

“…Remark 5. 4 The charged Riemannian Penrose inequality with corners established in [6] is presented in dimension 3, following [22]. However, as illustrated in the appendix of [21], the argument holds up to dimension 7 (where the standard problem concerning the regularity of minimal surfaces prevents it immediately being generalised to higher dimensions than that).…”