CONTENTS OF PART I OF VOLUME TWO xxi this energy. The nonexistence of steady and expanding breathers on closed manifolds, originally proved by one of the authors, may be proved using the energy functional and an associated invariant. We also discuss the classical entropy and its relation to Perelman's energy. Chapter 6. In this chapter we discuss Perelman's remarkable entropy functional. This functional is actually an energy-entropy quantity which combines Perelman's energy with the classical entropy using a positive pa- rameter r which, in the context of Ricci flow, plays the dual roles of the scale and minus time. We compute the first variation of the entropy and derive its monotonicity under the Ricci flow coupled to the adjoint heat equation. The entropy formula, via the consideration of test functions concentrated at points, leads to a volume noncollapsing result for all solutions to the Ricci flow on closed manifolds, called no local collapsing. This result also yields a strong injectivity radius estimate and rules out the formation of the cigar soliton as a finite time singularity model on closed manifolds. By minimizing the entropy functional over all functions satisfying a con- straint and then minimizing over all scales r, we obtain two geometric in- variants, one depending on the metric and scale and one depending only on the metric. The consideration of these invariants is useful in proving the nonexistence of nontrivial shrinking breather solutions on closed manifolds. We also provide an improved version of the no local collapsing theorem, also due to Perelman. Our presentation is based on the diameter bound result of Topping, whose proof we also sketch. Finally we discuss variational formulas for the modified scalar curva- ture, which is an integrand for Perelman's entropy functional, the second variation of energy and entropy functionals, and also a matrix Harnack-type calculation for the adjoint heat equation coupled to the Ricci flow. Chapter 7. In this chapter we give a detailed introduction to Perel- man's reduced distance, also called the ^-function. To reduce technicalities, we first consider the analogous function corresponding to fixed Riemannian metrics, which is simply the function d (p, q) /4r. In the Ricci flow case we first consider the ^-length. After presenting some basic properties of the ^-length, we compute its first variation formula and discuss the existence of £-geodesics, which are the critical points of the £-length. Associated to the ^-length is the L-distance, which is obtained by taking the infimum of the ^-length over paths with given endpoints. We compute the first space- and time-derivatives of the L-distance. This is partially analogous to the Gauss lemma in Riemannian geometry. Next we compute the second variation formula for the £-length, and motivated by space-time considerations, we express the formula in terms of Hamilton's matrix Harnack quadratic. This second variation formula yields an estimate for the Hessian of the L-distance (and hence for the reduced distance). Next we derive a number of differential equalities and inequalities for the reduced distance. These inequalities are the basis for the use of the reduced distance in the study of singularity formation under
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