Pointwise lower scalar curvature bounds for \(C^0\) metrics via regularizing Ricci flow
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In this paper we propose a class of local definitions of weak lower scalar curvature bounds that is well defined for \(C^0\) metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from \(C^0\) initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from \(C^0\) initial data.
I would like to thank my advisor, Richard Bamler, for introducing me to this project, and for his help and encouragement. I would also like to thank Christina Sormani, for posing the problem of torus rigidity for Definition 1.2 to me, and for showing relevant references to me. Finally, I would like to thank Chao Li, for showing the work of Simon [Sim02] to me, and for many helpful comments on a previous version of this paper. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE 1752814. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
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