On the Geometry and Uniqueness of Asymptotically Locally Hyperbolic Static Vacuum Black Holes

Abstract We establish several geometric characterizations and rigidity results for 3-dimensional asymptotically locally hyperbolic (ALH) static spaces with horizon boundary. Notably, a 3-dimensional ALH static space with toroidal infinity and strictly non-spherical horizons is isometric to a toroidal Kottler metric. Furthermore, we show that the surface gravity of a static horizon with spherical infinity is bounded below by 3 3, with equality achieved only by the critical AdS-Schwarzschild metric. Consequently, Poincaré-Einstein fillings of S^2 S^1 () S 2 × S 1 (λ) arising from these spaces have length parameter (3) ^-1 λ ≤ (3) - 1, which supports a recent conjecture of Chang-Yang-Zhang18. Finally, static horizons with hyperbolic infinity and non-negative Chruściel-Herzlich mass obey the reverse Riemannian Penrose inequality. In conjunction with work of Ge-Wang-Wu-Xia 29, we use this fact to obtain uniqueness of static ALH graphs with hyperbolic infinities. These results follow from a generalization of the Minkowski inequality in AdS-Schwarzschild space due to Brendle-Hung-Wang 15. Using optimal coefficients for the sub-static Heintze-Karcher inequality from 24, we construct a new monotone quantity under inverse mean curvature flow (IMCF) in static spaces with negative cosmological constant. Another fundamental tool developed in this paper is a regularity theorem for IMCF in ALH manifolds. Specifically, we prove that a weak solution of IMCF in an ALH 3-manifold with horizon boundary is eventually smooth. This extends the regularity theorem for a spherical infinity due to Shi-Zhu 52.