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

TL;DR

This paper extends the stability results of the positive mass theorem from asymptotically flat to asymptotically hyperbolic graphs, showing that small perturbations in mass imply geometric closeness to the hyperbolic space.

Contribution

It adapts existing stability techniques to the hyperbolic setting, establishing a similar stability result for asymptotically hyperbolic graphs with scalar curvature bounds.

Findings

Stability of the positive mass theorem for asymptotically hyperbolic graphs.

Extension of techniques from flat to hyperbolic geometries.

Quantitative estimates relating mass to geometric closeness.

Abstract

The positive mass theorem states that the total mass of a complete asymptotically flat manifold with non-negative scalar curvature is non-negative; moreover, the total mass equals zero if and only if the manifold is isometric to the Euclidean space. Huang and Lee [2015] proved the stability of the Positive Mass Theorem for a class of $n$-dimensional ($n \geq 3$) asymptotically flat graphs with non-negative scalar curvature, in the sense of currents. Motivated by their work and using results of Dahl, Gicquaud and Sakovich [2013], we adapt their ideas to obtain a similar result regarding the stability of the positive mass theorem, in the sense of currents, for a class of $n$-dimensional $(n \geq 3)$ asymptotically hyperbolic graphs with scalar curvature bigger than or equal to $-n(n-1)$.