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A positive mass theorem for low-regularity Riemannian metrics

Grant, JDE and Tassotti, N (2014) A positive mass theorem for low-regularity Riemannian metrics

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Abstract

We show that the positive mass theorem holds for continuous Riemannian metrics that lie in the Sobolev space $W^{2, n/2}_{loc}$ for manifolds of dimension less than or equal to $7$ or spin-manifolds of any dimension. More generally, we give a (negative) lower bound on the ADM mass of metrics for which the scalar curvature fails to be non-negative, where the negative part has compact support and sufficiently small $L^{n/2}$ norm. We show that a Riemannian metric in $W^{2, p}_{loc}$ for some $p > \frac{n}{2}$ with non-negative scalar curvature in the distributional sense can be approximated locally uniformly by smooth metrics with non-negative scalar curvature. For continuous metrics in $W^{2, n/2}_{loc}$, there exist smooth approximating metrics with non-negative scalar curvature that converge in $L^p_{loc}$ for all $p < \infty$.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
AuthorsEmailORCID
Grant, JDEUNSPECIFIEDUNSPECIFIED
Tassotti, NUNSPECIFIEDUNSPECIFIED
Date : 27 August 2014
Related URLs :
Additional Information : This is an arXiv publication
Depositing User : Symplectic Elements
Date Deposited : 04 Nov 2014 16:49
Last Modified : 29 May 2015 01:33
URI: http://epubs.surrey.ac.uk/id/eprint/806437

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