Rectifiability of Singular Sets in Lower Ricci Curvature
Abstract：In this talk, we will consider the Gromov-Hausdorff limit space (X,d) of a sequence of n-manifolds with lower Ricci curvature bound and noncollapsed volume. The limit space has a singular-regular decomposition X=R\cup S and dim S<=n-2 proved by Cheeger-Colding. In this talk we will study the structure of the singular set S and show that the singular set is (n-2)-rectifiable. We will also discuss the quantitative estimate of the singular set. The proofs are based on some new estimates on neck regions and a decomposition theorem which covers a general ball by neck regions and good balls. Our main focus in this talk is the proof of the decomposition theorem. This is a joint work with Jeff Cheeger and Aaron Naber.