On large volume preserving stable cmc surfaces in initial data sets

Michael Eichmair ETH Jan Metzger Universit¨at Potsdam

Differential Geometry mathscidoc:1609.10265

Journal of Differential Geometry, 91, (1), 81-102, 2012
Let (M, g) be a complete 3-dimensional asymptotically flat manifold with everywhere positive scalar curvature. We prove that, given a compact subset K  M, all volume preserving stable constant mean curvature surfaces of sufficiently large area will avoid K. This complements the results of G. Huisken and S.-T. Yau [17] and of J. Qing and G. Tian [26] on the uniqueness of large volume preserving stable constant mean curvature spheres in initial data sets that are asymptotically close to Schwarzschild with mass m > 0. The analysis in [17] and [26] takes place in the asymptotic regime of M. Here we adapt ideas from the minimal surface proof of the positive mass theorem [32] by R. Schoen and S.-T. Yau and develop geometric properties of volume preserving stable constant mean curvature surfaces to handle surfaces that run through the part of M that is far from Euclidean.
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@inproceedings{michael2012on,
  title={ON LARGE VOLUME PRESERVING STABLE CMC SURFACES IN INITIAL DATA SETS},
  author={Michael Eichmair, and Jan Metzger},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913225609445673927},
  booktitle={Journal of Differential Geometry},
  volume={91},
  number={1},
  pages={81-102},
  year={2012},
}
Michael Eichmair, and Jan Metzger. ON LARGE VOLUME PRESERVING STABLE CMC SURFACES IN INITIAL DATA SETS. 2012. Vol. 91. In Journal of Differential Geometry. pp.81-102. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913225609445673927.
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