Regularity of manifolds with bounded Ricci curvature and the codimension $4$ conjecture

Friday, October 3, 2014 -
2:00pm to 3:00pm
This talk will concern joint work with Aaron Naber on the regularity of noncollapsed Riemannian manifolds $M^n$ with bounded Ricci curvature $|{\rm Ric}_{M^n}|\leq n-1$, as well as their Gromov-Hausdorff limit spaces, $(M^n_j,d_j)\stackrel{d_{GH}}{\longrightarrow}(X,d)$, where $d_j$ denotes the Riemannian distance. We will explain a proof of the conjecture that $X$ is smooth away from a closed subset of  codimension $4$. By combining this with the ideas of quantitative stratification, we obtain a priori $L^q$ estimates on the full curvature tensor, for all $q<2$.  We also prove a conjecture of Anderson stating that for all $v>0$, $ <\infty$, the collection of $4$-manifolds $(M^4,g)$ with $|{\rm Ric}_{M^n}|\leq 3$, ${\rm Vol}(M^3)\geq v$, ${\rm diam}(M^4)\leq d$, contains a most a finite number of diffeomorphism types. A local version of this is used to how that noncollapsed $4$-manifolds with bounded Ricci curvature have a priori $L^2$ Riemannian curvature estimates. 
Jeff Cheeger
Event Location: 
Fine Hall 314