Abstract
We prove a sharp Alexandrov–Fenchel-type inequality for star-shaped, strictly mean convex hypersurfaces in hyperbolic n-space, n ≥ 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 ≥ 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, Chruściel and Simon on the validity of a Penrose-type inequality for exotic black holes.
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Communicated by James A. Isenberg.
L. L. de Lima was partially supported by a CNPq/Brazil research grant.
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de Lima, L.L., Girão, F. An Alexandrov–Fenchel-Type Inequality in Hyperbolic Space with an Application to a Penrose Inequality. Ann. Henri Poincaré 17, 979–1002 (2016). https://doi.org/10.1007/s00023-015-0414-0
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DOI: https://doi.org/10.1007/s00023-015-0414-0