A positive mass theorem for Lipschitz metrics with small singular sets

2011/10/28 by Lee, Dan A. · 1 citation
#53C20 #83C99 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1110.6485

Abstract

We prove that the positive mass theorem applies to Lipschitz metrics as long as the singular set is low-dimensional, with no other conditions on the singular set. More precisely, let g be an asymptotically flat Lipschitz metric on a smooth manifold Mⁿ, such that n<8 or M is spin. As long as g has bounded C² norm and nonnegative scalar curvature on the complement of some singular set S of Minkowski dimension less than n/2, the mass of g must be nonnegative. We conjecture that the dimension of S need only be less than n-1 for the result to hold. These results complement and contrast with earlier results of H. Bray, P. Miao, and Y. Shi and L.-F. Tam, where S is a hypersurface.

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