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

Levi Lopes de Lima, Frederico Girão

Published 2012-09-03, updated 2014-06-06Version 4

We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequality for star-shaped, strictly mean convex hypersurfaces in hyperbolic $n$-space, $n\geq 3$. 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.