Contents of Part III of Volume Two Well, you know, we’re doing what we can. From “Revolution” by The Beatles Chapter 17. Perelman’s entropy W leads to the µ-invariant. We dis- cuss qualitative properties of the µ-invariant such as lower and upper bounds and we give a proof of the fact that limτ→0+ µ (g, τ) = 0. We also discuss applications of the µ-invariant monotonicity formula. This includes the re- cent classification by Z.-L. Zhang of compact finite time singularity models as shrinking gradient Ricci solitons. We revisit the proof of the existence of a smooth minimizer for W, providing more details than in Part I, and we also show that when the isometry group acts transitively, the minimizer is not unique for sufficiently small τ. Related to renormalization group con- siderations, some low-loop calculations are presented. Chapter 18. We discuss some tools used in the study of the Ricci flow including the changing distances estimate for solutions of Ricci flow, point picking methods, rough monotonicity of the size of necks in complete noncompact manifolds with positive sectional curvature, and a local form of the weakened no local collapsing theorem. Chapter 19. With the goal of understanding compactness in higher dimensions, we introduce the notion of ‘κ-solution with Harnack’, which is a variant of Perelman’s notion of κ-solution. In dimensions 2 and 3 we show that κ-solutions with Harnack must have bounded curvature. We also discuss the construction of Perelman’s rotationally symmetric ancient solution on Sn, the result that κ-solutions with Harnack must have bounded curvature, the existence of an asymptotic shrinker in a κ-solution (correcting a gap (no pun intended) in Part I), and the κ-gap theorem. Chapter 20. We show that noncompact κ-solutions have asymptotic scalar curvature ratio ASCR = and asymptotic volume ratio AVR = 0 the latter result does not require the κ-noncollapsed at all scales assump- tion. We show that solutions which are almost ancient and have bounded nonnegative curvature operator are collapsed at large scales and we obtain a curvature estimate in noncollapsed balls. We prove that the collection of κ-solutions with Harnack is compact modulo scaling. In dimension 3 this is equivalent to Perelman’s compactness theorem and implies scaled derivative of curvature estimates. xiii
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