English

The positive mass theorem for manifolds with distributional curvature

General Relativity and Quantum Cosmology 2014-08-20 v1 Analysis of PDEs

Abstract

We formulate and prove a positive mass theorem for n-dimensional spin manifolds whose metrics have only the Sobolev regularity C0W1,nC^0 \cap W^{1,n}. At this level of regularity, the curvature of the metric is defined in the distributional sense only, and we propose here a (generalized) notion of ADM mass for such a metric. Our main theorem establishes that if the manifold is asymptotically flat and has non-negative scalar curvature distribution, then its (generalized) ADM mass is well-defined and non-negative, and vanishes only if the manifold is isometric to Euclidian space. Prior applications of Witten's spinor method by Lee and Parker and by Bartnik required the much stronger regularity W2,2W^{2,2}. Our proof is a generalization of Witten's arguments, in which we must treat the Dirac operator and its associated Lichnerowicz-Weitzenbock identity in the distributional sense and cope with certain averages of first-order derivatives of the metric over annuli that approach infinity. Finally, we observe that our arguments are not specific to scalar curvature and also allow us to establish a universal positive mass theorem.

Keywords

Cite

@article{arxiv.1408.4431,
  title  = {The positive mass theorem for manifolds with distributional curvature},
  author = {Dan A. Lee and Philippe G. LeFloch},
  journal= {arXiv preprint arXiv:1408.4431},
  year   = {2014}
}

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26 pages