Asymptotically hyperbolic manifolds with small mass

TL;DR

This paper investigates asymptotically hyperbolic manifolds with small mass, showing that the conformal factor approaches one as the mass approaches zero, thus supporting the positive mass theorem in this setting.

Contribution

It demonstrates that for certain asymptotically hyperbolic manifolds, the conformal factor converges to one as the mass diminishes, providing insight into the structure near zero mass.

Findings

Conformal factor tends to one as mass tends to zero.

Supports the positive mass theorem for conformally hyperbolic manifolds.

Analyzes manifolds conformally hyperbolic outside a fixed radius.

Abstract

For asymptotically hyperbolic manifolds of dimension $n$ with scalar curvature at least equal to $-n(n-1)$ the conjectured positive mass theorem states that the mass is non-negative, and vanishes only if the manifold is isometric to hyperbolic space. In this paper we study asymptotically hyperbolic manifolds which are also conformally hyperbolic outside a ball of fixed radius, and for which the positive mass theorem holds. For such manifolds we show that the conformal factor tends to one as the mass tends to zero.