Preface I didn’t have time to write you a short letter, so I wrote you a long one instead. – Samuel Clemens What Part III is about I’m taking the time for a number of things That weren’t important yesterday. – From “Fixing a Hole” by The Beatles This is Part III (a.k.a. ∆Rijk), a sequel to Part I ([40] a.k.a. Rijk) and Part II ([41] a.k.a. ∂ ∂t Rijk) of this volume (Volume Two) on techniques and applications of the Ricci flow (we shall refer to Volume One ([42] a.k.a. gij) as Volume One). In Part I we discussed various geometric topics in Ricci flow such as Ricci solitons, an introduction to the K¨ ahler–Ricci flow, Hamilton’s Cheeger– Gromov-type compactness theorem, Perelman’s energy and entropy mono- tonicity, the foundations of Perelman’s reduced distance function, the re- duced volume, applications to the analysis of ancient solutions, and a primer on 3-manifold topology. In Part II we discussed mostly analytic topics in Ricci flow including weak and strong maximum principles for scalar heat-type equations and systems on compact and noncompact manifolds, B¨ ohm and Wilking’s clas- sification of closed manifolds with 2-positive curvature operator, Shi’s local derivative estimates, Hamilton’s matrix estimate, and Perelman’s estimate for fundamental solutions of the adjoint heat equation. Here, in Part III, we discuss mostly geometric-analytic topics in Ricci flow. In particular, we discuss properties of Perelman’s entropy functional, point picking methods, aspects of Perelman’s theory of κ-solutions including the κ-gap theorem, compactness theorem, and derivative estimates, Perel- man’s pseudolocality theorem, and aspects of the heat equation with respect to static and evolving metrics related to Ricci flow. In the appendices we review metric and Riemannian geometry including the space of points at infinity and Sharafutdinov retraction for complete noncompact manifolds with nonnegative sectional curvature. As in previous volumes, we have en- deavored, as much as possible, to make the chapters independent of each other. ix

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