The Penrose Inequality and Positive Mass Theorem with Charge for Manifolds with Asymptotically Cylindrical Ends
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Annales Henri Poincar´ e
The Penrose Inequality and Positive Mass Theorem with Charge for Manifolds with Asymptotically Cylindrical Ends Jaroslaw S. Jaracz Abstract. We establish the charged Penrose inequality 1 q2 m≥ ρ+ 2 ρ for time-symmetric initial data sets having an outermost minimal surface boundary and finitely many asymptotically cylindrical ends, with an appropriate rigidity statement. This is accomplished by a doubling argument based on the work of Weinstein and Yamada (Commun Math Phys 257:703–723, 2005). arXiv:math/0405602) and a subsequent application of the ordinary charged Penrose inequality as established by Khuri et al. (Contemp Math 653:219–226, 2015. arXiv:1308.3771; J Differ Geom 106:451–498, 2017. arXiv:1409.3271). Furthermore, the techniques used in the aforementioned proof allow for an alternative proof of the positive mass theorem with charge for such manifolds, a result originally obtained in Bartnik and Chrusciel (Journal f¨ ur die reine und angewandte Mathematik 579:13–73, 2005).
1. Introduction One of the most famous inequalities of mathematical general relativity is the Penrose inequality. Conjectured by Penrose [18] in the early 1970s using a heuristic argument based on the establishment viewpoint of gravitational collapse and the assumption of cosmic censorship, the Penrose inequality relates the mass m to the surface area A of the black hole. Defining the area radius ρ by A = 4πρ2 , the Penrose inequality takes the form m≥
1 ρ. 2
(1.1)
J. S. Jaracz
Ann. Henri Poincar´e
The Riemannian Penrose inequality is a special case, where the mass m is the ADM mass of an asymptotically flat 3-manifold having nonnegative scalar curvature and A is the area of the outermost minimal surface (with possibly multiple components). It was proved in the late 1990s by Huisken and Ilmanen using weak inverse mean curvature flow (IMCF) with the area A being the largest connected component of the outermost minimal surface [12] and in full generality by Bray using a novel conformal flow of metrics [5]. In the case that the outermost minimal surface has a single boundary component, inequality (1.1) can be extended to include charge yielding q2 1 ρ+ , (1.2) m≥ 2 ρ where q is the total charge enclosed by the apparent horizon. One might then conjecture that (1.2) holds when the outermost apparent horizon has multiple components. However, a time-symmetric, asymptotically flat counterexample was constructed in [21] by gluing two copies of the Majumdar–Papapetrou initial data sets. It is important to point out that this does not provide a counterexample to cosmic censorship. As pointed out by Jang [13], (1.2) is equivalent to two inequalities, (1.3) m − m2 − q 2 ≤ ρ ≤ m + m2 − q 2 and only the upper bound follows from Penrose’s heuristic arguments. The counterexample violates the lower bound. The usual geometry associated with the Penrose inequality is that of a manifold with boundary, where the boundary is a compact outermost apparent horizon (which coincides with the an outermost minimal surface in the t
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