Lp-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. To cite this article: J. Cheeger, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 195–198.