V l l l PREFACE model is a solution of the flow that arises as the limit of a sequence of dila- tions of an original solution approaching a singularity.) Since the underlying manifolds of such limit solutions are topologically simple, a detailed analy- sis of the singularities which arise in dimension three is therefore possible. For example, showing that certain singularity models are shrinking cylin- ders is one of the cornerstones for enabling geometric-topological surgeries to be performed on singular solutions to the Ricci flow on 3-manifolds. The Li-Yau-Hamilton-type differential Harnack quantity is another innovation this yields an a priori estimate for a certain expression involving the cur- vature and its first and second derivatives in space. This estimate allows one to compare a solution at different points and times. In particular, it shows that in the presence of a nonnegative curvature operator, the scalar curvature does not decrease too fast. Another consequence of Hamilton's differential Harnack estimate is that slowly forming singularities (those in which the curvature of the original solution grows more quickly than the parabolically natural rate) lead to singularity models that are stationary solutions. The convergence theory of Cheeger and Gromov has had fundamental consequences in Riemannian geometry. In the setting of the Ricci flow, Gromov's compactness theorem may be improved to obtain C°° convergence of a sequence of solutions to a smooth limit solution. For singular solutions, Perelman's recent No Local Collapsing Theorem allows one to dilate about sequences of points and times approaching the singularity time in such a way that one can obtain a limit solution that exists infinitely far back in time. The analysis of such limit solutions is important in Hamilton's singularity theory. A guide for the reader The reader of this book is assumed to have a basic knowledge of Rie- mannian geometry. Familiarity with algebraic topology and with nonlinear second-order partial differential equations would also be helpful but is not strictly necessary. In Chapters 1 and 2, we begin the study of the Ricci flow by considering special solutions which exhibit typical properties of the Ricci flow and which guide our intuition. In the case of an initial homogeneous metric, the solution remains homogenous so that its analysis reduces to the study of a system of ODE. We study examples of such solutions in Chapter 1. Solutions which exist on long time intervals such as those which exist since time oo or until time +oo are very special and can appear as singularity models. In Chapter 2, we both present such solutions explicitly and provide some intuition for how they arise. An important example of a stationary solution is the so- called "cigar soliton" on R2 intuitive solutions include the neckpinch and degenerate neckpinch. In Chapter 2, we discuss degenerate neckpinches
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