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

Recent progress on geometric analysis and Riemannian geometry October 21, 2024 - October 25, 2024

October 21, 2024 (03:30 PM PDT - 04:30 PM PDT)
Speaker(s): Paula Burkhardt-Guim (Stony Brook University)
Location: SLMath: Eisenbud Auditorium, Online/Virtual
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
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Abstract

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We show that any $L^\infty$ Riemannian metric $g$ on $\R^n$ 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.

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