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

We show that any $L^{\mathrm{\infty }}$ 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 $L^{\mathrm{\infty }}$ to the Euclidean metric. The approximation is given by time slices of the Ricci-DeTurck flow, which converge locally in $C^{\mathrm{\infty }}$ to $g$ away from the singular set. We also identify conditions under which a smooth Ricci-DeTurck flow starting from a $L^{\mathrm{\infty }}$ 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.