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

TL;DR

This paper proves a sharp geometric inequality in hyperbolic space and applies it to establish an optimal Penrose inequality for asymptotically hyperbolic graphs, advancing understanding of black hole physics in negative cosmological constant settings.

Contribution

It introduces two new monotone quantities along inverse mean curvature flow and derives an optimal Penrose inequality in hyperbolic space, confirming a conjecture for time-symmetric space-times.

Findings

Established a sharp Alexandrov-Fenchel-type inequality in hyperbolic space.

Proved an optimal Penrose inequality for asymptotically hyperbolic graphs with minimal horizons.

Extended results to asymptotically locally hyperbolic graphs with higher genus horizons.

Abstract

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