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Geometric Analysis Seminar

Convergence and regularity theorems for entropy and scalar curvature lower bounds

Robin Neumayer (Carnegie Mellon University)

Location:  zoom
Date & time: Tuesday, 05 October 2021 at 2:50AM - 3:50PM

Abstract: In this talk, we consider Riemannian manifolds with almost non-negative scalar curvature and Perelman entropy. We establish an epsilon-regularity theorem showing that such a space must be close to Euclidean space in a suitable sense. Interestingly, such a result is false with respect to the Gromov-Hausdorff and Intrinsic Flat distances, and more generally the metric space structure is not controlled under entropy and scalar lower bounds. Instead, we introduce the notion of the \(d_p\) distance between (in particular) Riemannian manifolds, which measures the distance between \(W^{1,p}\) Sobolev spaces, and it is with respect to this distance that the \(\epsilon\) regularity theorem holds. We will discuss various applications to manifolds with scalar curvature and entropy lower bounds. This is joint work with Man-Chun Lee and Aaron Naber.

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