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

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 ($n\ge 3$) 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 $(n\ge 3)$ asymptotically hyperbolic graphs with scalar curvature bigger than or equal to $-n(n-1)$.

23 pages, 1 figure. This is a pre-print of an article published in Annals of Global Analysis and Geometry. The final authenticated version is available online at: https://doi.org/10.1007/s10455-019-09674-9