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.

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