On the stability of the positive mass theorem for asymptotically hyperbolic graphs
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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 \(-\,n(n-1)\).
KeywordsAsymptotically hyperbolic graphs Stability of hyperbolic positive mass theorem Asymptotically hyperbolic manifolds
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.
- 1.Allen, B.: IMCF and the stability of the PMT and RPI under \(L^2\) convergence. Ann. Henri Poincaré 19, 1–24 (2017)Google Scholar
- 7.Chruściel, P.T., Delay, E.: The hyperbolic positive energy theorem. arXiv:1901.05263v2 (2019)
- 18.Herzlich, M.: Mass formulae for asymptotically hyperbolic manifolds, AdS/CFT correspondence: Einstein metrics and their conformal boundaries, IRMA Lect. Math. Theor. Phys., vol. 8, Eur. Math. Soc., Zürich, pp. 103–121 (2005)Google Scholar
- 19.Huang, L.-H., Jang, H.-C., Martin, D.: Mass rigidity for hyperbolic manifolds. arXiv:1904.12010v1 (2019)
- 23.Lam, M.-K.G.: The graph cases of the riemannian positive mass and penrose inequalities in all dimensions. arXiv:1010.4256v1 (2010)
- 27.Montiel, S., Ros, A.: Compact hypersurfaces: the Alexandrov theorem for higher order mean curvatures. In: Differential Geometry, Pitman Monogr. Surveys Pure Appl. Math., vol. 52, Longman Sci. Tech., Harlow, pp. 279–296 (1991)Google Scholar
- 31.Schoen, R., Yau, S.-T.: Positive scalar curvature and minimal hypersurface singularities. arXiv:1704.05490v1 (2017)
- 32.Simon, L.: Lectures on geometric measure theory. In: Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3. Australian National University, Centre for Mathematical Analysis, Canberra (1983)Google Scholar