The ADM mass of asymptotically flat hypersurfaces

TL;DR

This paper derives integral formulas for the ADM mass of asymptotically flat hypersurfaces in warped product manifolds, extending previous results and establishing new inequalities and notions of quasi-local mass.

Contribution

It generalizes Lam's Euclidean graph results to broader warped product settings and introduces a novel link between ADM mass and the Newton tensor in the presence of Killing fields.

Findings

New integral formulas for ADM mass in warped product manifolds

Validation of positive mass and Penrose inequalities in new classes of manifolds

Discussion of a quasi-local mass concept in this geometric context

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 infinity, thus extending Lam's recent results on Euclidean graphs to this broader context. As applications we exhibit, in any dimension, new classes of manifolds for which versions of the Positive Mass and Riemannian Penrose inequalities hold and discuss a notion of quasi-local 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.