Smoothing L^∞ Riemannian metrics with nonnegative scalar curvature outside of a singular set

2024/06/07 by Burkhardt-Guim, Paula · 1 citation
#53C21 #53E20 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2406.04564

Abstract

We show that any L^∞ Riemannian metric g on ℝⁿ that is smooth with nonnegative scalar curvature away from a singular set of finite (n-α)-dimensional Minkowski content, for some α>2, admits an approximation by smooth Riemannian metrics with nonnegative scalar curvature, provided that g is sufficiently close in L^∞ to the Euclidean metric. The approximation is given by time slices of the Ricci-DeTurck flow, which converge locally in C^∞ to g away from the singular set. We also identify conditions under which a smooth Ricci-DeTurck flow starting from a L^∞ 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.

Cited by

• Ricci-DeTurck flow of almost continuous L²-metrics, and metrics with distributional scalar curvature bounded from below

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