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

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Abstract

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

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Acknowledgements

I would like to thank Lan-Hsuan Huang for pointing out this problem to me, for all her support and motivating discussions. I would also like to thank Carla Cederbaum, Kwok-Kun Kwong and Jason Ledwidge for very valuable comments and discussions, and to Anna Sakovich for a thorough reading of the first draft of this work and for her very helpful observations. In addition, I would like to thank the Carl Zeiss Foundation for the generous support. This work was partially funded by NSF Grant DMS 1452477.

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  1. Department of Mathematics, Universität Tübingen, 72076, Tübingen, Germany

    Armando J. Cabrera Pacheco

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Correspondence to Armando J. Cabrera Pacheco.

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Cabrera Pacheco, A.J. On the stability of the positive mass theorem for asymptotically hyperbolic graphs. Ann Glob Anal Geom 56, 443–463 (2019). https://doi.org/10.1007/s10455-019-09674-9

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