An Alexandrov-Fenchel-type inequality in hyperbolic space with an application to a Penrose inequality

Abstract

We prove a sharp Alexandrov-Fenchel-type inequality for star-shaped, strictly mean convex hypersurfaces in hyperbolic $n$-space, $n\ge 3$. The argument uses two new monotone quantities along the inverse mean curvature flow. As an application we establish, in any dimension, an optimal Penrose inequality for asymptotically hyperbolic graphs carrying a minimal horizon, with the equality occurring if and only if the graph is an anti-de Sitter-Schwarzschild solution. This sharpens previous results by Dahl-Gicquaud-Sakovich and settles, for this class of initial data sets, the conjectured Penrose inequality for time-symmetric space-times with negative cosmological constant. We also explain how our methods can be easily adapted to derive an optimal Penrose inequality for asymptotically locally hyperbolic graphs in any dimension $n\ge 3$. When the horizon has the topology of a compact surface of genus at least one, this provides an affirmative answer, for this class of initial data sets, to a question posed by Gibbons, Chrusciel and Simon on the validity of a Penrose-type inequality for exotic black holes.

21 pages; no figures; published version, which incorporates the main results in arXiv:1304.7887

References in corpus (2)

• Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds
• Present status of the Penrose inequality

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