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The entropy formula for the Ricci flow and its geometric applications Grisha Perelman∗ February 1, 2008 Introduction 1. The Ricci flow equation, introduced by Richard Hamilton [H 1], is the evolution equation d dtgij (t) = −2Rij for a riemannian metric gij (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 Rmt = △Rm + Q, where Q is a certain quadratic expression of the curvatures. In particular, the scalar curvature R satisfies Rt = △R + 2|Ric| 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 ∗St.Petersburg branch of Steklov Mathematical Institute, Fontanka 27, St.Petersburg 191011, Russia. Email: perelman@pdmi.ras.ru or perelman@math.sunysb.edu ; I was partially supported by personal savings accumulated during my visits to the Courant Institute in the Fall of 1992, to the SUNY at Stony Brook in the Spring of 1993, and to the UC at Berkeley as a Miller Fellow in 1993-95. I’d like to thank everyone who worked to make those opportunities available to me. 1 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 (−∞, 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,§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 → ∞, and be nonsingular, in the sense that the normalized curvatures 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 2 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 → ∞ 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 K¨ahler-Ricci flow on K¨ahler manifolds with positive first Chern class; these will be discussed in a separate paper.

 

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