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On scalar curvature rigidity of vacuum static spaces

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Abstract

In this paper we extend the local scalar curvature rigidity result in Brendle and Marques (J Differ Geom 88:379–394, 2011) to a small domain on general vacuum static spaces, which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the paper (Corvino, Commun Math Phys 214:137–189, 2000). We obtain the local scalar curvature rigidity of bounded domains in hyperbolic spaces. We also obtain the global scalar curvature rigidity for conformal deformations of metrics in the domains, where the lapse functions are positive, on vacuum static spaces with positive scalar curvature, and show such domains are maximal, which generalizes the work in Hang and Wang (Commun Anal Geom 14:91–106, 2006).

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Notes

  1. After we posted the previous version of this paper on arXiv, we were informed that the rigidity part of Theorem 1.3 had been known in [3].

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Acknowledgments

We would like to thank Professors Justin Corvino, Pengzi Miao and Carla Cerderbaum for their interests and stimulating discussions.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Santa Cruz, CA, 95064, United States

    Jie Qing

  2. School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou, 510275, Guangdong, China

    Wei Yuan

Authors
  1. Jie Qing
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  2. Wei Yuan
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Correspondence to Wei Yuan.

Additional information

Research of the authors supported by NSF Grant DMS-1005295 and DMS-1303543.

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Qing, J., Yuan, W. On scalar curvature rigidity of vacuum static spaces. Math. Ann. 365, 1257–1277 (2016). https://doi.org/10.1007/s00208-015-1302-0

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  • DOI: https://doi.org/10.1007/s00208-015-1302-0

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