x PREFACE aspects of singularity formation in dimension 3. In this volume, we continue the study of the fundamental properties of the Ricci flow with particular emphasis on their application to the study of singularities. We pay particu- lar attention to dimension 3, where we describe some aspects of Hamilton's and Perelman's nearly complete classification of the possible singularities.2 As we saw in Volume One, Ricci solitons (i.e., self-similar solutions), dif- ferential Harnack inequalities, derivative estimates, compactness theorems, maximum principles, and injectivity radius estimates play an important role in the study of the Ricci flow. The maximum principle was used extensively in the 3-dimensional results we presented. Some of the other techniques were presented only in the context of the Ricci flow on surfaces. In this volume we take a more detailed look at these general topics and also describe some of the fundamental new tools of Perelman which almost complete Hamilton's partial classification of singularities in dimension 3. In particular, we discuss Perelman's energy, entropy, reduced distance, and some applications. Much of Perelman's work is independent of dimension and leads to a new under- standing of singularities. It is difficult to overemphasize the importance of the reduced distance function, which is a space-time distance-like function (not necessarily nonnegative!) which is intimately tied to the geometry of solutions of the Ricci flow and the understanding of forming singularities. We also discuss stability and the linearized Ricci flow. Here the emphasis is not just on one solution to the Ricci flow, but on the dependence of the so- lutions on their initial conditions. We hope that this direction of study may have applications to showing that certain singularity types are not generic. This volume is divided into two parts plus appendices. For the most part, the division is along the lines of whether the techniques are geometric or analytic. However, this distinction is rather arbitrary since the techniques in Ricci flow are often a synthesis of geometry and analysis. The first part is intended as an introduction to some basic geometric techniques used in the study of the singularity formation in general dimensions. Particular atten- tion is paid to finite time singularities on closed manifolds, where the spatial maximum of the curvature tends to infinity in finite time. We also discuss some basic 3-manifold topology and reconcile this with some classification results for 3-dimensional finite time singularities. The partial classification of such singularities is used in defining Ricci flow with surgery. In particular, given a good enough understanding of the singularities which can occur in dimension 3, one can perform topological-geometric surgeries on solutions to the Ricci flow either right before or at the singularity time. One would then like to continue the solution to the Ricci flow until the next singularity and iterate this process. In the end one hopes to infer the existence of a geometric decomposition on the underlying 3-manifold. This is what Hamil- ton's program aims to accomplish and this is the same framework on which Not all singularity models have been classified, even for finite time solutions of the Ricci flow on closed 3-manifolds. Apparently this is independent of Hamilton's program for Thurston geometrization.
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