A definition of the mass aspect function for weakly regular asymptotically hyperbolic manifolds

TL;DR

This paper introduces a unified, low-regularity definition of the mass aspect function for asymptotically hyperbolic manifolds, connecting existing approaches and demonstrating its covariance and computability via Ricci tensor.

Contribution

It unifies two existing notions of mass for asymptotically hyperbolic manifolds by defining a broad, low-regularity mass aspect function with covariance properties.

Findings

The mass aspect function can be computed using the Ricci tensor.

The new definition applies to very low regularity metrics.

The approach demonstrates asymptotic rigidity of hyperbolic space.

Abstract

In contrast to the well-known and unambiguous notion of ADM mass for asymptotically Euclidean manifolds, the notion of mass for asymptotically hyperbolic manifolds admits several interpretations. Historically, there are two approaches to defining the mass in the asymptotically hyperbolic setting: the mass aspect function of Wang defined on the conformal boundary at infinity, and the mass functional of Chru\'sciel and Herzlich which may be thought of as the closest asymptotically hyperbolic analogue of the ADM mass. In this paper we unify these two approaches by introducing an ADM-style definition of the mass aspect function that applies to a broad range of asymptotics and in very low regularity. Additionally, we show that the mass aspect function can be computed using the Ricci tensor. Finally, we demonstrate that this function exhibits favorable covariance properties under changes of charts at infinity, which includes a proof of the asymptotic rigidity of hyperbolic space in the context of weakly regular metrics.