A positive mass theorem for low-regularity Riemannian metrics

We show that the positive mass theorem holds for continuous Riemannian metrics that lie in the Sobolev space $W_{loc}^{2,n\mathrm{/}2}$ 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 $L^{n\mathrm{/}2}$ norm. We show that a Riemannian metric in $W_{loc}^{2,p}$ 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 $W_{loc}^{2,n\mathrm{/}2}$, there exist smooth approximating metrics with non-negative scalar curvature that converge in $L_{loc}^p$ for all $p<\mathrm{\infty }$.