A positive mass theorem for Lipschitz metrics with small singular sets

Lee, Dan

doi:10.1090/S0002-9939-2013-11871-X

A positive mass theorem for Lipschitz metrics with small singular sets
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by Dan A. Lee

Proc. Amer. Math. Soc. 141 (2013), 3997-4004

DOI: https://doi.org/10.1090/S0002-9939-2013-11871-X

Published electronically: July 30, 2013

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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 $M^n$ such that $n<8$ or $M$ is spin. As long as $g$ has bounded $C^2$-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 earlier results of H. Bray, P. Miao, and Y. Shi and L.-F. Tam where $S$ is a hypersurface.

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