The ADM mass of asymptotically flat hypersurfaces

Abstract

We provide integral formulae for the ADM mass of asymptotically flat hypersurfaces in Riemannian manifolds with a certain warped product structure in a neighborhood of 'spatial' infinity, thus extending Lam's recent results on Euclidean graphs to this broader context. As applications we exhibit, in any dimension, new examples of manifolds for which versions of the Positive Mass and Riemannian Penrose inequalities hold and discuss a notion of quasilocal mass in this setting. The proof explores a novel connection between the co-vector defining the ADM mass of a hypersurface as above and the Newton tensor associated to its shape operator, which takes place in the presence of an ambient Killing field.

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

“…The inequality (1.2) has recently become relevant in the context of the Penrose inequality for asymptotically flat graphs carrying a minimal horizon [L] [dLG1] and for asymptotically hyperbolic graphs carrying a constant mean curvature horizon [dLG2]. As an application of Theorem 1.1 we establish an optimal Penrose inequality for asymptotically hyperbolic graphs carrying a minimal horizon, including the rigidity statement according to which the equality holds only if (M, g) is the graph realization of an anti-de Sitter-Schwarzschild solution; see Theorem 1.2.…”

A Penrose inequality for graphs over Kottler space

“…Recently Lam [30] gave an elegant proof of (1.1) for asymptotically flat graphs over R n for all dimensions. His proof was later extended in [14,29,35]. Very recently, a general Penrose inequality for a higher order mass was conjectured in [21], which is true for the graph cases [21,32] and conformally flat cases [22].…”

An Alexandrov–Fenchel-type inequality for hypersurfaces in the sphere

“…Remark 1. The warped product structure (7) and the identities (11) and (12) are satisfied, for example, by the de Sitter-Schwarzchild manifold.…”