Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass

In this work, we study the positive mass theorem under critical low regularity assumptions using Ricci flow smoothing. We show that asymptotically flat manifolds $(M^n,g)$ of regularity $L^{\infty }\cap W^{1,n}$ with non-negative distributional scalar curvature have non-negative ADM mass. Furthermore, when the ADM mass vanishes, the manifold is globally isometric to Euclidean space with respect to an integral distance introduced by De Cecco-Palmieri. This extends the recent work of Hafemann to the critical regularity case. Our approach is based on showing that Riemannian metrics of regularity $L^{\infty }\cap W^{1,n}$, whose scalar curvature is bounded from below in the distributional sense, admit a Ricci flow smoothing whose scalar curvature is bounded from below by the same initial lower bound in the classical sense. In contrast, Cecchini-Frenck-Zeidler constructed examples of metrics which are in $L^{\infty }\cap W^{1,p}$ for all $2<p<n$, and whose distributional scalar curvature is bounded from below, that cannot be approximated by smooth metrics with the same scalar curvature lower bound. In this sense, our result is optimal.