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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A positive mass theorem for Lipschitz metrics with small singular sets


Author: Dan A. Lee
Journal: Proc. Amer. Math. Soc. 141 (2013), 3997-4004
MSC (2010): Primary 53C20, 83C99
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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Additional Information

Dan A. Lee
Affiliation: CUNY Graduate Center and Queens College, 365 Fifth Avenue, New York, New York 10016
Email: dan.lee@qc.cuny.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11871-X
PII: S 0002-9939(2013)11871-X
Received by editor(s): December 14, 2011
Published electronically: July 30, 2013
Additional Notes: This work was partially supported by NSF DMS No. 0903467 and a PSC CUNY Research Grant.
Communicated by: Lei Ni
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.