Tobias H. Colding

Lower bounds on Ricci curvature and the almost rigidity of warped products

The basic rigidity theorems for manifolds of nonnegative or positive Ricci curvature are the volume cone implies metric cone theorem, the maximal diameter theorem, [Cg], and the splitting theorem, [CG]. Each asserts that if a certain geometric quantity (volume or diameter) is as large as possible relative to the pertinent lower bound on Ricci curvature, then the metric on the manifold in question is a warped product metric of a particular type. In this paper we provide quantitative generalizations of the above mentioned results. Among the applications are the splitting theorem for GromovHausdorff limit spaces X, where Mn -* X, Ricmn ? -i see [FY]. Other applications include the assertion that for complete manifolds, M, with Ricmf > 0 and Euclidean volume growth, all tangent cones at infinity are metric cones; compare [BKN], [CT], [P1]. Via resealing arguments, there are also strong consequences for the local structure of manifolds whose Ricci curvature satisfies a fixed lower bound and for their Gromov-Hausdorff limits. Some of these are announced in [CCol]; for a more detailed discussion see [CCo2], [CCo3], [CCo4]. Our work further develops and significantly extends techniques which were introduced in [Col], [Co2] and significantly extended in [Co3], in order to prove certain stability conjectures of Anderson-Cheeger, Gromov and Perelman. The results of [Col]-[Co3] were announced in [Co4]. We briefly review some of those results. Let dGH denote the Gromov-Hausdorff distance between metric spaces; see [GLP]. Let S denote the unit sphere and recall that S is the unique complete

Ricci curvature and volume convergence

The purpose of this paper is to give a new (integral) estimate of distances and angles on manifolds with a given lower Ricci curvature bound. This will provide us with an integral version of the Toponogov comparison triangle theorem for Ricci curvature and almost extreme triangles (see the earlier works [Cl] and [C2] for an analog of this when the manifold has positive Ricci curvature). Using this, we prove the Anderson-Cheeger conjecture saying that the volume is a continuous function on the space of all closed n-manifolds with Ricci curvature greater or equal to -(n - 1) equipped with the GromovHausdorff metric. We also prove Gromovs conjecture (for n 57 3) saying that an almost nonnegatively Ricci curved n-manifold with first Betti number equal to n is a torus. Further, we prove a conjecture of Anderson-Cheeger saying that an open n-manifold with nonnegative Ricci curvature whose tangent cone at infinity is in is itself in. Finally we prove a conjecture of Fukaya-Yamaguchi. We will now describe these results in more detail. Let dGH denote the Gromov-Hausdorff distance [GLP]. First we have the following result which was conjectured by Anderson-Cheeger.

On the singularities of spaces with bounded Ricci curvature

Abstract. ((Without Abstract)).n

Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman

We show that the Ricci flow becomes extinct in finite time on any Riemannian 3-manifold without aspherical summands. In this note we prove some bounds for the extinction time for the Ricci flow on certain 3-manifolds. Our interest in this comes from a question that Grisha Perelman asked the first author at a dinner in New York City on April 25th of 2003. His question was what happens to the Ricci flow on the 3-sphere when one starts with an arbitrary metric? In particular, does the flow become extinct in finite time? He then went on to say that one of the difficulties in answering this is that he knew of no good way of constructing minimal surfaces for such a metric in general. However, there is a natural way of constructing such surfaces and that comes from the min-max argument where the minimal of all maximal slices of sweep-outs is a minimal surface; see, for instance, (2). The idea is then to look at how the area of this min-max surface changes under the flow. Geometrically the area measures a kind of width of the 3-manifold and as we will see for certain 3-manifolds (those, like the 3-sphere, whose prime decomposition contains no aspherical factors) the area becomes zero in finite time corresponding to that the solution becomes extinct in finite time. Moreover, we will discuss a possible lower bound for how fast the area becomes zero. Very recently Perelman posted a paper (see (9)) answering his original question about finite extinction time. However, even after the appearance of his paper, then we still think that our slightly different approach may be of interest. In part because it is in some ways geometrically more natural, in part because it also indicates that lower bounds should hold, and in part because it avoids using the curve shortening flow that he simultaneously with the Ricci flow needed to invoke and thus our approach is in some respects technically easier. Let M 3 be a smooth closed orientable 3-manifold and let g(t) be a one-parameter family of metrics on M evolving by the Ricci flow, so

Smooth compactness of self-shrinkers

We prove a smooth compactness theorem for the space of embedded self-shrinkers in

Lower Bounds for Nodal Sets of Eigenfunctions

RR^3

Embedded minimal disks: Proper versus nonproper—global versus local

. Since self-shrinkers model singularities in mean curvature flow, this theorem can be thought of as a compactness result for the space of all singularities and it plays an important role in studying generic mean curvature flow.

Rigidity of generic singularities of mean curvature flow

We prove lower bounds for the Hausdorff measure of nodal sets of eigenfunctions.

WIDTH AND MEAN CURVATURE FLOW

We construct a sequence of (compact) embedded minimal disks in a ball in R 3 with boundaries in the boundary of the ball and where the curvatures blow up only at the center. The sequence converges to a limit which is not smooth and not proper. If instead the sequence of embedded disks had boundaries in a sequence of balls with radii tending to infinity, then we have shown previously that any limit must be smooth and proper.

Constraints on singularities under Ricci curvature bounds

Shrinkers are special solutions of mean curvature flow (MCF) that evolve by rescaling and model the singularities. While there are infinitely many in each dimension, Colding and Minicozzixa0II (Ann. Math. 175(2):755–833, 2012) showed that the only generic are round cylinders Sk×Rn−k. We prove here that round cylinders are rigid in a very strong sense. Namely, any other shrinker that is sufficiently close to one of them on a large, but compact, set must itself be a round cylinder.To our knowledge, this is the first general rigidity theorem for singularities of a nonlinear geometric flow. We expect that the techniques and ideas developed here have applications to other flows.Our results hold in all dimensions and do not require any a priori smoothness.