Geometric Analysis and Topology Seminar

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Convergence and regularity theorems for entropy and scalar curvature lower bounds

Speaker: Robin Neumayer, Carnegie Mellon

Location: Online

Videoconference link: https://nyu.zoom.us/j/94747124588

Date: Wednesday, April 13, 2022, 11 a.m.

Synopsis:

 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. This is joint work with Man-Chun Lee and Aaron Naber.