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

The positive mass theorem states that the total mass of a complete asymptotically flat manifold with nonnegative scalar curvature is nonnegative; moreover, the total mass equals zero if and only if the manifold is isometric to the Euclidean space. Huang and Lee (Commun Math Phys 337(1):151–169, 2015) proved the stability of the positive mass theorem for a class of n-dimensional (n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 3$$\end{document}) asymptotically flat graphs with nonnegative scalar curvature, in the sense of currents. Motivated by their work and using results of Dahl et al. (Ann Henri Poincaré 14(5):1135–1168, 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≥3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n \ge 3)$$\end{document} asymptotically hyperbolic graphs with scalar curvature bigger than or equal to -n(n-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\,n(n-1)$$\end{document}.