Penrose Type Inequalities for Asymptotically Hyperbolic Graphs

In this paper, we study asymptotically hyperbolic manifolds given as graphs of asymptotically constant functions over hyperbolic space ${\mathbb{H}^n}$ . The graphs are considered as unbounded hypersurfaces of ${\mathbb{H}^{n+1}}$ which carry the induced metric and have an interior boundary. For such manifolds, the scalar curvature appears in the divergence of a 1-form involving the integrand for the asymptotically hyperbolic mass. Integrating this divergence, we estimate the mass by an integral over the inner boundary. In case the inner boundary satisfies a convexity condition, this can in turn be estimated in terms of the area of the inner boundary. The resulting estimates are similar to the conjectured Penrose inequality for asymptotically hyperbolic manifolds. The work presented here is inspired by Lam’s article (The graph cases of the Riemannian positive mass and Penrose inequalities in all dimensions. http://arxiv.org/abs/1010.4256, 2010) concerning the asymptotically Euclidean case. Using ideas developed by Huang and Wu (The equality case of the penrose inequality for asymptotically flat graphs. http://arxiv.org/abs/1205.2061, 2012), we can in certain cases prove that equality is only attained for the anti-de Sitter Schwarzschild metric.