L_p-bounds on curvature, elliptic estimates and rectifiability of singular sets

We announce results on rectifiability of singular sets of pointed metric spaces which are pointed Gromov-Hausdorff limits on sequences of Riemannian manifolds, satisfying uniform lower bounds on Ricci curvature and volume, and uniform L_p-bounds on curvature. The rectifiability theorems depend on estimates for Hess_{h L2p}, ( ∇Hess_h. Hess_h ^p-2)L₂, where Δh = c, for some constant c. We also observe that (absent any integral bound on curvature) in the Kähler case, given a uniform 2-sided bound on Ricci curvature, the singular set has complex codimension 2.