Penrose Type Inequalities for Asymptotically Hyperbolic Graphs

Abstract

In this paper we study asymptotically hyperbolic manifolds given as graphs of asymptotically constant functions over hyperbolic space H n . The graphs are considered as unbounded hypersurfaces of H n+1 which carry the induced metric and have an interior boundary. For such manifolds the scalar curvature appears in the divergence of a 1-form involving the integrand for the asymptotically hyperbolic mass. Integrating this divergence we estimate the mass by an integral over the inner boundary. In case the inner boundary satisfies a convexity condition this can in turn be estimated in terms of the area of the inner boundary. The resulting estimates are similar to the conjectured Penrose inequality for asymptotically hyperbolic manifolds. The work presented here is inspired by Lam's article [22] concerning the asymptotically Euclidean case. Using ideas developed by Huang and Wu in [19] we can in certain cases prove that equality is only attained for the anti-de Sitter Schwarzschild metric. appeared on arXiv. In the first of these papers an Alexandrov-Fenchel type inequality for hypersurfaces in hyperbolic space is stated, which together with Proposition 4.1 implies the Penrose inequality (1) for graphs. Certain steps of the proof seem to need further clarification, for example the convergence of hypersurfaces to round spheres under the inverse mean curvature flow. However, combining with arguments of the second paper [7] the result should follow. Note also that a special case of [7, Theorem 2] follows from our formula (13) in Section 4.2.

On the stability of the positive mass theorem for asymptotically hyperbolic graphs

“…As a consequence of the level sets being star-shaped, Theorem 1.1 only works when the initial data set contains a horizon with a single connected component (or no horizon at all). However, this seems to be consistent with the existing result for the Riemannian Penrose inequality in this setting (see [10,12]). The case when the horizon has several connected components will not be discussed here.…”

“…Namely, in Section 2 we provide all the necessary definitions and terminology. In Section 3 we obtain results similar to those in [10] in the case when the background metric of H n+1 is rescaled in a special way; this rescaling is used to obtain a differential inequality for the volume of the level sets as in [17]. In Section 4 we obtain the desired differential inequality and use it to estimate the maximum height of asymptotically hyperbolic graphs.…”

“…Asymptotically hyperbolic manifolds are, roughly speaking, Riemannian manifolds whose metrics approach the metric b at infinity. As in the asymptotically flat case, one is interested in a global invariant referred to as total mass (in the asymptotically flat case this notion is known as the ADM mass); however, in the asymptotically hyperbolic setting, this notion is more subtle and depends on the structure of the set N = {V ∈ C ∞ (H n ) | Hess b V = V b} (see [8,23,10]). Note that V (0) := V (0) (r) = cosh r and V (i) := x i sinh r, where x i denote the coordinate functions on R n restricted to S n−1 , give a basis of the vector space N .…”

“…It is worth to notice that Huang and Lee in [17] had to treat differently the cases n = 3 and n = 4, mainly because of the use of the Minkowski inequality in R n . Interestingly, in our setting, using a Minkowski-like inequality obtained by Montiel and Ros in [24] and used by Dahl, Gicquaud and Sakovich in [10], the argument works for any dimension n ≥ 3.…”

“…We will use the notation in [10] and [17] as much as possible. In addition, the structure of the article is similar to [17].…”