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

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

2006-02-15

arxiv.org/abs/math/0602321v1

Mathematics

Differential Geometry

34 pages

Scientific paper

In \cite{ly, ly2}, 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 \cite{st}. 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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