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

TL;DR

This paper demonstrates how to approximate certain $L^ fty$ Riemannian metrics with nonnegative scalar curvature outside a singular set using Ricci-DeTurck flow, ensuring smooth convergence away from singularities.

Contribution

It introduces a method to smooth $L^ fty$ metrics with controlled scalar curvature outside singular sets via Ricci-DeTurck flow, extending scalar curvature preservation results.

Findings

Approximation of $L^ fty$ metrics by smooth metrics with nonnegative scalar curvature.

Identification of conditions for scalar curvature nonnegativity under Ricci-DeTurck flow.

Convergence of flow to the original metric outside singularities.

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.