##### Department of Mathematics,

University of California San Diego

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### Math 258 - Differential Geometry Seminar

## Robin Neumayer

#### Northwestern

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

##### 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, 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.

Host: Luca Spolaor

### February 24, 2021

### 10:15 AM

### Zoom link: Meeting ID: 988 8132 1752

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