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.

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