Abstract
The rigidity of the positive mass theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We prove a corresponding stability theorem for spaces that can be realized as graphical hypersurfaces in \({\mathbb{R}^{n+1}}\). Specifically, for an asymptotically flat graphical hypersurface \({M^n\subset \mathbb{R}^{n+1}}\) of nonnegative scalar curvature (satisfying certain technical conditions), there is a horizontal hyperplane \({\Pi\subset \mathbb{R}^{n+1}}\) such that the flat distance between M and \({\Pi}\) in any ball of radius \({\rho}\) can be bounded purely in terms of n, \({\rho}\), and the mass of M. In particular, this means that if the masses of a sequence of such graphs approach zero, then the sequence weakly converges (in the sense of currents, after a suitable vertical normalization) to a flat plane in \({\mathbb{R}^{n+1}}\). This result generalizes some of the earlier findings of Lee and Sormani (J Reine Angew Math 686:187–220, 2014) and provides some evidence for a conjecture stated there.
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Communicated by P. T. Chruściel
L.-H. Huang was partially supported by the NSF through DMS-1301645 and DMS-1308837. This material is also based upon work supported by the NSF under Grant No. 0932078 000, while L.-H. Huang and D. A. Lee were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2013 program in Mathematical General Relativity.
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Huang, LH., Lee, D.A. Stability of the Positive Mass Theorem for Graphical Hypersurfaces of Euclidean Space. Commun. Math. Phys. 337, 151–169 (2015). https://doi.org/10.1007/s00220-014-2265-9
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DOI: https://doi.org/10.1007/s00220-014-2265-9