A positive mass theorem for low-regularity Riemannian metrics

2014/08/27 by Grant, James D. E., Tassotti, Nathalie · 1 citation
#53C20 #58J05 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1408.6425

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 > (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 < ∞.

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