An Alexandrov–Fenchel-Type Inequality in Hyperbolic Space with an Application to a Penrose Inequality

Abstract

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 for 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 spacetimes with negative cosmological constant.

A Minkowski Inequality for Hypersurfaces in the Anti‐de Sitter‐Schwarzschild Manifold

“…We note that, after this paper was written, several related inequalities for hypersurfaces in hyperbolic space have appeared in the literature; see, e.g., [19,24]. 1 2 Star-Shaped Hypersurfaces in the AdS-Schwarzschild Manifold LEMMA 2.1.…”

Inverse mean curvature flows in the hyperbolic 3-space revisited

“…A new strategy relying on the conformal structure and the Sobolev type inequality on the sphere at infinity was devised in [1,2] to tackle this difficulty and to obtain Penrose-Gibbon inequalities in the asymptotically flat case. Such a strategy has been further developed by several authors [5,8] to solve related problems.…”

Deforming the Scalar Curvature of the De Sitter–Schwarzschild Space

“…In the asymptotically flat case (ǫ = 0), the (exterior) Schwarzschild space, which is the corresponding KID, is rigid if 3 ≤ n ≤ 7, a result that follows from the sharp Penrose inequality established for spin manifolds in these dimensions by Bray-Lee [BrL], following previous contributions by Bray [Br] and HuiskenIlmanen [HI]. Thus, the putative rigidity of Schwarzschild space remains unsettled in dimension n ≥ 8, except in the graphical category, where it follows from the work of Lam [L] and Huang-Wu [HW]; see also [dLG1]. In the asymptotically hyperbolic case (ǫ = −1), the corresponding KID is the anti-de Sitter-Schwarzschild space, which is only known to be rigid in two cases.…”

“…First, rigidity holds in dimension n = 3, a consequence of an estimate for a certain normalized volume due to Brendle-Chodosh [BC]; for sufficiently small deformations, this also follows from the work of Ambrozio [Am], who established a sharp Penrose inequality in this setting. Also, it holds in the graphical case for any dimension, as confirmed by the optimal Penrose inequality proved in [dLG2]. Thus, the important question of whether the anti-de Sitter-Schwarzschild space is rigid remains wide open if n ≥ 4.…”