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Abstract: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.

Submission history

From: Man Chun Lee [view email]
[v1] Mon, 22 Jun 2026 12:50:32 UTC (27 KB)