$d_p$ Convergence and $\varepsilon$-regularity theorems for entropy and scalar curvature lower bounds

Robin Neumayer (Northwestern)

16-Mar-2021, 13:45-14:45 (3 years ago)

Abstract: In this talk, we consider Riemannian manifolds with almost non-negative scalar curvature and Perelman entropy. We establish an $\varepsilon$-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, including a compactness and limit structure theorem for sequences, a uniform $L^\infty$ Sobolev embedding, and a priori $L^p$ scalar curvature bounds for $p<1$. This is joint work with Man-Chun Lee and Aaron Naber.

differential geometry

Audience: researchers in the topic


B.O.W.L Geometry Seminar

Organizers: Joel Fine, Lorenzo Foscolo*, Peter Topping
Curators: Jason D Lotay*, Costante Bellettini, Bruno Premoselli, Felix Schulze, Huy The Nguyen, Marco Guaraco, Michael Singer
*contact for this listing

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