A positive mass theorem for low-regularity Riemannian metrics

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We show that the positive mass theorem holds for continuous Riemannian metrics that lie in the Sobolev space W², ⁿ/²loc 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 Lⁿ/² norm. We show that a Riemannian metric in W², ᵖloc 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 W², ⁿ/²loc, there exist smooth approximating metrics with non-negative scalar curvature that converge in Lᵖloc for all p < ∞.