Smoothing $L^{\infty }$ Riemannian metrics with nonnegative scalar curvature outside of a singular set

Abstract

We show that any $L^{\infty }$ Riemannian metric $g$ on ${\mathbb{R}}^n$ that is smooth with nonnegative scalar curvature away from a singular set of finite $(n-\alpha )$-dimensional Minkowski content, for some $\alpha >2$, admits an approximation by smooth Riemannian metrics with nonnegative scalar curvature, provided that $g$ is sufficiently close in $L^{\infty }$ to the Euclidean metric. The approximation is given by time slices of the Ricci-DeTurck flow, which converge locally in $C^{\infty }$ to $g$ away from the singular set. We also identify conditions under which a smooth Ricci-DeTurck flow starting from a $L^{\infty }$ metric that is uniformly bilipschitz to Euclidean space and smooth with nonnegative scalar curvature away from a finite set of points must have nonnegative scalar curvature for positive times.

v2: 29 pages. Special treatment given to the case of the 0-dimensional Minkowski content (Theorem 1.5 in this version) without the epsilon-bilipschitz assumption, discussion of Question 3 updated with reference to work that has recently appeared in the literature, slightly strengthened intermediary results, introduction rearranged, abstract updated, contact information updated, reference fixed