xxii CONTENTS OF PART I OF VOLUME TWO the Ricci flow. We also consider the reduced distance in the special cases of Einstein solutions, and more generally, gradient Ricci solitons. There is a whole space-time geometry associated to the £-length and re- duced distance. We consider the notions of C- Jacobi fields and /^-exponential map. We derive properties of these objects including the C- Jacobi equation, bounds for £-Jacobi fields, the £-cut locus, and £-Jacobian. We derive bounds for the reduced distance, its spatial gradient, and its time-derivative. Since the reduced distance is a Lipschitz function, we recall the basic prop- erties of Lipschitz functions and formulate the precise sense in which differ- ential inequalities for the reduced distance hold. Chapter 8. We discuss applications of the study of the reduced distance to the study of finite time singularities for the Ricci flow. First we consider the reduced volume associated to a static metric. This is simply the integral of the transplanted Euclidean heat kernel using the exponential map based at some point. In the case of nonnegative Ricci curvature, the static metric reduced volume is monotonically nonincreasing. This corresponds to the fact that the reduced volume integrand is a weak subsolution to the heat equation. With analogies to no local collapsing in mind, we relate the static metric reduced volume to volume ratios of balls. Next we consider Perelman's reduced volume for the Ricci flow. For all solutions of the Ricci flow on closed manifolds, the reduced volume is monotonically nondecreasing. We present various heuristic proofs and then justify these proofs using the basic properties of the reduced distance as a Lipschitz function and the £-Jacobian developed in the previous chapter. We prove a weakened version of the no local collapsing theorem using the reduced volume monotonicity. This proof is somewhat technical since one needs some estimates for the ^-exponential map. Its advantage over the entropy proof given in Chapter 6 is that it holds for complete solutions of the Ricci flow on noncompact manifolds with bounded curvature. Perelman's no local collapsing theorem tells us that singularity models in dimension 3 are ancient ^-solutions. To obtain more canonical limits, one often needs to take backward limits in time and rescale to obtain new ancient ^-solutions. The reduced distance function may be used to show that certain backward limits of ancient Ac-solutions are shrinking gradient Ricci solitons. In dimension 3 such a soliton must either be a spherical space form, the cylinder S2 x R, or its Z2-quotient. This has important consequences for singularity formation in dimension 3. Chapter 9. In this chapter we give a survey of some of the basic 3- manifold topology which is related to the Ricci flow. Appendix A. We review the contents of Volume One and other aspects of basic Riemannian geometry and Ricci flow. Appendix B. In many cases, solutions to the Ricci flow limit to Ricci solitons. We present some low-dimensional results of this type for certain
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