Stability of Euclidean 3-space for the positive mass theorem
Abstract
We show that the Euclidean 3-space R3 is stable for the Positive Mass Theorem in the following sense. Let (Mi,gi) be a sequence of complete asymptotically flat 3-manifolds with nonnegative scalar curvature and suppose that the ADM mass m(gi) of one end of Mi converges to 0. Then for all i, there is a subset Zi in Mi such that Mi∖Zi contains the given end, the area of the boundary ∂Zi converges to zero, and (Mi∖Zi,gi) converges to R3 in the pointed measured Gromov-Hausdorff topology for any choice of basepoints. This confirms a conjecture of G. Huisken and T. Ilmanen. Additionally, we find an almost quadratic upper bound for the area of ∂Zi in terms of m(gi). As an application of the main result, we also prove R. Bartnik’s strict positivity conjecture.
Copyright and License
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.
Acknowledgement
We would like to thank Gerhard Huisken for suggesting an application of the main theorem to the Bartnik capacity, and Marcus Khuri, Hubert Bray, Christos Mantoulidis and Hyun Chul Jang for helpful discussions. The writing of this paper was substantially improved thanks to suggestions of the referees.
Funding
C.D. would like to thank his advisor Xiuxiong Chen for his encouragements. Part of the revised version was supported by the NSF grant DMS-1928930, while C.D. was in residence at the Simons Laufer Mathematical Sciences Institute (formerly MSRI) in Berkeley, California, during the Fall 2024 semester. A.S. was partially supported by the NSF grant DMS-2104254. This research was conducted during the period A.S. served as a Clay Research Fellow.
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Additional details
- National Science Foundation
- DMS-1928930
- National Science Foundation
- DMS-2104254
- Accepted
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2024-11-25Accepted
- Available
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2024-12-04Published online
- Publication Status
- Published