Communications in Analysis and Geometry

Volume 15 (2007)

Number 2

A generalization of Liu-Yau's quasi-local mass

Pages: 249 – 282



Mu-Tao Wang

Shing-Tung Yau


In Positivity of quasi-local mass (C.-C. M. Liu, S.-T. Yau, Phys. Rev. Lett. 90(23) (2003), 231102, 4) and Positivity of quasi-local mass II (C.-C. M. Liu, S.-T. Yau, J. Amer. Math. Soc. 19(1) (2006), 181-204), Liu and the second author propose a definition of the quasi-local mass and prove its positivity. This is demonstrated through an inequality which in turn can be interpreted as a total mean curvature comparison theorem for isometric embeddings of a surface of positive Gaussian curvature. The Riemannian version corresponds to an earlier theorem of Shi and Tam (Positive mass theorem and the boundary behavior of compact manifolds with nonnegative scalar curvature, Y. Shi, L.-F. Tam, J. Differential Geom. 62(1) (2002), 79-125). In this article, we generalize such an inequality to the case when the Gaussian curvature of the surface is allowed to be negative. This is done by an isometric embedding into the hyperboloid in the Minkowski space and a future-directed time-like quasi-local energy-momentum is obtained.

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