# Seminars & Colloquia Calendar

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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