dp Convergence and ϵ-regularity theorems for entropy and scalar curvature lower bounds

In this talk, we consider Riemannian manifolds with almost non-negative scalar curvature and Perelman entropy. We establish an ϵ-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 dp distance between (in particular) Riemannian manifolds, which measures the distance between W1,p Sobolev spaces, and it is with respect to this distance that the ϵ 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 and a priori Lp scalar curvature bounds for p<1 This is joint work with Man-Chun Lee and Aaron Naber.