The Minkowski Formula and the Quasi-Local Mass

  • Po-Ning ChenEmail author
  • Mu-Tao Wang
  • Shing-Tung Yau


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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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, RiversideRiversideUSA
  2. 2.Department of MathematicsColumbia UniversityNew YorkUSA
  3. 3.Department of MathematicsHarvard UniversityCambridgeUSA

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