The spacetime positive mass theorem in dimensions less than eight

Lan-Hsuan Huang University of Connecticut Dan A. Lee Queens College and CUNY Graduate Center Richard Schoen University of California Michael Eichmair University of Vienna

Differential Geometry mathscidoc:1803.10001

Distinguished Paper Award in 2018

Journal of European Mathematical Society, 18, 83-121, 2016
We prove the spacetime positive mass theorem in dimensions less than eight. This theorem asserts that for any asymptotically flat initial data set that satisfies the dominant energy condition, the inequality E>=|P| holds, where (E, P) is the ADM energy-momentum vector. Previously, this theorem was only known for spin manifolds [38]. Our approach is a modification of the minimal hypersurface technique that was used by the last named author and S.-T. Yau to establish the time-symmetric case of this theorem [30, 27]. Instead of minimal hypersurfaces, we use marginally outer trapped hypersurfaces (MOTS) whose existence is guaranteed by earlier work of the first named author [14]. An important part of our proof is to introduce an appropriate substitute for the area functional that is used in the time-symmetric case to single out certain minimal hypersurfaces. We also establish a density theorem of independent interest and use it to reduce the general case of the spacetime positive mass theorem to the special case of initial data that has harmonic asymptotics and satisfies the strict dominant energy condition.
Positive mass theorem, marginally outer trapped surfaces
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@inproceedings{lan-hsuan2016the,
  title={The spacetime positive mass theorem in dimensions less than eight},
  author={Lan-Hsuan Huang, Dan A. Lee, Richard Schoen, and Michael Eichmair},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180314112731040217979},
  booktitle={Journal of European Mathematical Society},
  volume={18},
  pages={83-121},
  year={2016},
}
Lan-Hsuan Huang, Dan A. Lee, Richard Schoen, and Michael Eichmair. The spacetime positive mass theorem in dimensions less than eight. 2016. Vol. 18. In Journal of European Mathematical Society. pp.83-121. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180314112731040217979.
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