Introduction

1 1   The Ricci flow equation, introduced by Richard Hamilton [H 1], is the evolution equation $\frac{d}{dt}g_{ij}(t)=-2R_{ij}$ for a riemannian metric $g_{ij}(t).$ In his seminal paper, Hamilton proved that this equation has a unique solution for a short time for an arbitrary (smooth) metric on a closed manifold. The evolution equation for the metric tensor implies the evolution equation for the curvature tensor of the form $Rm_t=\triangle Rm +Q,$ where $Q$ is a certain quadratic expression of the curvatures. In particular, the scalar curvature $R$ satisfies $R_t=\triangle R+2\vert$Ric$\vert^2,$ so by the maximum principle its minimum is non-decreasing along the flow. By developing a maximum principle for tensors, Hamilton [H 1,H 2] proved that Ricci flow preserves the positivity of the Ricci tensor in dimension three and of the curvature operator in all dimensions; moreover, the eigenvalues of the Ricci tensor in dimension three and of the curvature operator in dimension four are getting pinched pointwisely as the curvature is getting large. This observation allowed him to prove the convergence results: the evolving metrics (on a closed manifold) of positive Ricci curvature in dimension three, or positive curvature operator in dimension four converge, modulo scaling, to metrics of constant positive curvature.

Without assumptions on curvature the long time behavior of the metric evolving by Ricci flow may be more complicated. In particular, as $t$ approaches some finite time $T,$ the curvatures may become arbitrarily large in some region while staying bounded in its complement. In such a case, it is useful to look at the blow up of the solution for $t$ close to $T$ at a point where curvature is large (the time is scaled with the same factor as the metric tensor). Hamilton [H 9] proved a convergence theorem , which implies that a subsequence of such scalings smoothly converges (modulo diffeomorphisms) to a complete solution to the Ricci flow whenever the curvatures of the scaled metrics are uniformly bounded (on some time interval), and their injectivity radii at the origin are bounded away from zero; moreover, if the size of the scaled time interval goes to infinity, then the limit solution is ancient, that is defined on a time interval of the form $(-\infty , T).$ In general it may be hard to analyze an arbitrary ancient solution. However, Ivey [I] and Hamilton [H 4] proved that in dimension three, at the points where scalar curvature is large, the negative part of the curvature tensor is small compared to the scalar curvature, and therefore the blow-up limits have necessarily nonnegative sectional curvature. On the other hand, Hamilton [H 3] discovered a remarkable property of solutions with nonnegative curvature operator in arbitrary dimension, called a differential Harnack inequality, which allows, in particular, to compare the curvatures of the solution at different points and different times. These results lead Hamilton to certain conjectures on the structure of the blow-up limits in dimension three, see [H 4,$\S 26$]; the present work confirms them.

The most natural way of forming a singularity in finite time is by pinching an (almost) round cylindrical neck. In this case it is natural to make a surgery by cutting open the neck and gluing small caps to each of the boundaries, and then to continue running the Ricci flow. The exact procedure was described by Hamilton [H 5] in the case of four-manifolds, satisfying certain curvature assumptions. He also expressed the hope that a similar procedure would work in the three dimensional case, without any a priory assumptions, and that after finite number of surgeries, the Ricci flow would exist for all time $t\to\infty,$ and be nonsingular, in the sense that the normalized curvatures $\tilde{Rm}(x,t)=tRm(x,t)$ would stay bounded. The topology of such nonsingular solutions was described by Hamilton [H 6] to the extent sufficient to make sure that no counterexample to the Thurston geometrization conjecture can occur among them. Thus, the implementation of Hamilton program would imply the geometrization conjecture for closed three-manifolds.

In this paper we carry out some details of Hamilton program. The more technically complicated arguments, related to the surgery, will be discussed elsewhere. We have not been able to confirm Hamilton's hope that the solution that exists for all time $t\to\infty$ necessarily has bounded normalized curvature; still we are able to show that the region where this does not hold is locally collapsed with curvature bounded below; by our earlier (partly unpublished) work this is enough for topological conclusions.

Our present work has also some applications to the Hamilton-Tian conjecture concerning Kähler-Ricci flow on Kähler manifolds with positive first Chern class; these will be discussed in a separate paper.

2 2   The Ricci flow has also been discussed in quantum field theory, as an approximation to the renormalization group (RG) flow for the two-dimensional nonlinear $\sigma$-model, see [Gaw,$\S 3$] and references therein. While my background in quantum physics is insufficient to discuss this on a technical level, I would like to speculate on the Wilsonian picture of the RG flow.

In this picture, $t$ corresponds to the scale parameter; the larger is $t,$ the larger is the distance scale and the smaller is the energy scale; to compute something on a lower energy scale one has to average the contributions of the degrees of freedom, corresponding to the higher energy scale. In other words, decreasing of $t$ should correspond to looking at our Space through a microscope with higher resolution, where Space is now described not by some (riemannian or any other) metric, but by an hierarchy of riemannian metrics, connected by the Ricci flow equation. Note that we have a paradox here: the regions that appear to be far from each other at larger distance scale may become close at smaller distance scale; moreover, if we allow Ricci flow through singularities, the regions that are in different connected components at larger distance scale may become neighboring when viewed through microscope.

Anyway, this connection between the Ricci flow and the RG flow suggests that Ricci flow must be gradient-like; the present work confirms this expectation.

3 3   The paper is organized as follows. In $\S 1$ we explain why Ricci flow can be regarded as a gradient flow. In $\S 2,3$ we prove that Ricci flow, considered as a dynamical system on the space of riemannian metrics modulo diffeomorphisms and scaling, has no nontrivial periodic orbits. The easy (and known) case of metrics with negative minimum of scalar curvature is treated in $\S 2;$ the other case is dealt with in $\S 3,$ using our main monotonicity formula (3.4) and the Gaussian logarithmic Sobolev inequality, due to L.Gross. In $\S 4$ we apply our monotonicity formula to prove that for a smooth solution on a finite time interval, the injectivity radius at each point is controlled by the curvatures at nearby points. This result removes the major stumbling block in Hamilton's approach to geometrization. In $\S 5$ we give an interpretation of our monotonicity formula in terms of the entropy for certain canonical ensemble. In $\S 6$ we try to interpret the formal expressions , arising in the study of the Ricci flow, as the natural geometric quantities for a certain Riemannian manifold of potentially infinite dimension. The Bishop-Gromov relative volume comparison theorem for this particular manifold can in turn be interpreted as another monotonicity formula for the Ricci flow. This formula is rigorously proved in $\S 7;$ it may be more useful than the first one in local considerations. In $\S 8$ it is applied to obtain the injectivity radius control under somewhat different assumptions than in $\S 4.$ In $\S 9$ we consider one more way to localize the original monotonicity formula, this time using the differential Harnack inequality for the solutions of the conjugate heat equation, in the spirit of Li-Yau and Hamilton. The technique of $\S 9$ and the logarithmic Sobolev inequality are then used in $\S 10$ to show that Ricci flow can not quickly turn an almost euclidean region into a very curved one, no matter what happens far away. The results of sections 1 through 10 require no dimensional or curvature restrictions, and are not immediately related to Hamilton program for geometrization of three manifolds.

The work on details of this program starts in $\S 11,$ where we describe the ancient solutions with nonnegative curvature that may occur as blow-up limits of finite time singularities ( they must satisfy a certain noncollapsing assumption, which, in the interpretation of $\S 5,$ corresponds to having bounded entropy). Then in $\S 12$ we describe the regions of high curvature under the assumption of almost nonnegative curvature, which is guaranteed to hold by the Hamilton and Ivey result, mentioned above. We also prove, under the same assumption, some results on the control of the curvatures forward and backward in time in terms of the curvature and volume at a given time in a given ball. Finally, in $\S 13$ we give a brief sketch of the proof of geometrization conjecture.

The subsections marked by * contain historical remarks and references. See also [Cao-C] for a relatively recent survey on the Ricci flow.