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The Minkowski Formula and the Quasi-Local Mass

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Abstract

In this article, we estimate the quasi-local energy with reference to the Minkowski spacetime (Wang and Yau in Phys Rev Lett 102(2):021101, 2009; Commun Math Phys 288(3):919–942, 2009), the anti-de Sitter spacetime (Chen et al. in Commun Anal Geom, 2016. arXiv:1603.02975), or the Schwarzschild spacetime (Chen et al. in Adv Theor Math Phys 22(1):1–23, 2018). In each case, the reference spacetime admits a conformal Killing–Yano 2-form which facilitates the application of the Minkowski formula in Wang et al. (J Differ Geom 105(2):249–290, 2017) to estimate the quasi-local energy. As a consequence of the positive mass theorems in Liu and Yau (J Am Math Soc 19(1):181–204, 2006) and Shi and Tam (Class Quantum Gravity 24(9):2357–2366, 2007) and the above estimate, we obtain rigidity theorems which characterize the Minkowski spacetime and the hyperbolic space.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Riverside, Riverside, CA, 92521, USA

    Po-Ning Chen

  2. Department of Mathematics, Columbia University, New York, NY, 10027, USA

    Mu-Tao Wang

  3. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Shing-Tung Yau

Authors
  1. Po-Ning Chen
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  2. Mu-Tao Wang
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  3. Shing-Tung Yau
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Correspondence to Po-Ning Chen.

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Communicated by Mihalis Dafermos.

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P.-N. Chen is supported by NSF Grant DMS-1308164 and Simons Foundation collaboration Grant #584785, M.-T. Wang is supported by NSF Grants DMS-1105483 and DMS-1405152, and S.-T. Yau is supported by NSF Grants PHY-0714648 and DMS-1308244. This work was partially supported by a Grant from the Simons Foundation (#305519 to Mu-Tao Wang). Part of this work was carried out when P.-N. Chen and M.-T. Wang were visiting the Department of Mathematics and the Center of Mathematical Sciences and Applications at Harvard University.

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Chen, PN., Wang, MT. & Yau, ST. The Minkowski Formula and the Quasi-Local Mass. Ann. Henri Poincaré 20, 889–904 (2019). https://doi.org/10.1007/s00023-019-00766-7

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  • DOI: https://doi.org/10.1007/s00023-019-00766-7

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