A New Quasi-local Mass and Positivity

Abstract

We use an idea of Wang and Yau to give a new definition of quasi-local mass for a topological sphere in an initial date set. The new definition modifies Brown-York's definition by using certain spinor norm as lapse function. And it requires mean curvature of the topological sphere satisfies apparent horizon conditions, the topological sphere can be isometrically into Euclidean 3-space and mean curvature of the initial date set does not change sign. The positivity holds if we further assume the image of the topological sphere in Euclidean 3-space has nonnegative mean curvature.Comment: 13 pages, final version, the limiting case in Theorem 1 corrected, remark 1 and new references added, Acta Mathematica Sinica (English Series), to appea