We prove the following: Let (M, g, X) be a noncompact four dimensional shrinking soliton with bounded nonnegative curvature operator, then (M, g) is isometric to R 4 or a finite quotient of S 2 × R 2 or S 3 × R. In the process we also show that a complete shrinking soliton (M, g, X) with bounded curvature is gradient and κ-noncollapsed and the dilation of a Type I singularity is a shrinking soliton. Further in dimension three we show shrinking solitons with bounded curvature can be classified under only the assumption of Rc ≥ 0.

ABSTRACT. In this paper, we are concerned with the regularity of noncollapsed Riemannian manifolds (M n , g) with bounded Ricci curvature, as well as their Gromov-Hausdorff limit spaces (M, where d j denotes the Riemannian distance. Our main result is a solution to the codimension 4 conjecture, namely that X is smooth away from a closed subset of codimension 4. We combine this result with the ideas of quantitative stratification to prove a priori L q estimates on the full curvature |Rm| for all q < 2. In the case of Einstein manifolds, we improve this to estimates on the regularity scale. We apply this to prove a conjecture of

We prove a new estimate on manifolds with a lower Ricci bound which asserts that the geometry of balls centered on a minimizing geodesic can change in at most a Hölder continuous way along the geodesic. We give examples that show that the Hölder exponent, along with essentially all the other consequences that follow from this estimate, are sharp.Among the applications is that the regular set is convex for any noncollapsed limit of Einstein metrics. In the general case of a potentially collapsed limit of manifolds with just a lower Ricci curvature bound we show that the regular set is weakly convex and a.e. convex. We also show two conjectures of Cheeger-Colding. One of these asserts that the isometry group of any, even collapsed, limit of manifolds with a uniform lower Ricci curvature bound is a Lie group. The other asserts that the dimension of any limit space is the same everywhere.

Let Y n denote the Gromov-Hausdorff limit M n i d GH −→ Y n of v-noncollapsed riemannian manifolds with Ric M n i ≥ −(n − 1). The singular set S ⊂ Y has a stratification S 0 ⊂ S 1 ⊂ · · · ⊂ S, where y ∈ S k if no tangent cone at y splits off a factor R k+1 isometrically. Here, we define for all η > 0, 0 < r ≤ 1, the k-th effective singular stratum S k η,r satisfying η r S k η,r = S k . Sharpening the known Hausdorff dimension bound dim S k ≤ k, we prove that for all y, the volume of the r-tubular neighborhood of S k η,r satisfies Vol(T r (S k η,r ) ∩ B 1 2 (y)) ≤ c(n, v, η)r n−k−η . The proof involves a quantitative differentiation argument. This result has applications to Einstein manifolds. Let B r denote the set of points at which the C 2 -harmonic radius is ≤ r. If also the M n i are Kähler-Einstein with L 2 curvature bound, ||Rm|| L 2 ≤ C, then Vol(B r ∩ B 1 2 (y)) ≤ c(n, v, C)r 4 for all y. In the Kähler-Einstein case, without assuming any integral curvature bound on the M n i , we obtain a slightly weaker volume bound on B r which yields an a priori L p curvature bound for all p < 2.

ABSTRACT. In this paper we study the regularity of stationary and minimizing harmonic maps f : B 2 (p) ⊆ M → N between Riemannian manifolds. If S k ( f ) ≡ {x ∈ M : no tangent map at x is k + 1-symmetric} is the k th -stratum of the singular set of f , then it is well known that dim S k ≤ k, however little else about the structure of S k ( f ) is understood in any generality. Our first result is for a general stationary harmonic map, where weIn the case of minimizing harmonic maps we go further, and prove that the singular set S ( f ), which is well known to satisfy dim S ( f ) ≤ n − 3, is in fact n − 3-rectifiable with uniformly finite n − 3-measure. An effective version of this allows us to prove that |∇ f | has estimates in L Reifenberg, and give checkable criteria to determine when a set is k-rectifiable with uniform measure estimates.The new Reifenberg type theorems may be of some independent interest. The L 2 -subspace approximation theorem we prove is then used to help break down the quantitative stratifications into pieces which satisfy these criteria.