derivative estimates established in this chapter enable one to prove the long-
time existence theorem for the flow, which states that a unique solution to
the Ricci flow exists as long as its curvature remains bounded.
In Chapter 8, we begin the analysis of singularity models by discussing
the general procedures for obtaining limits of dilations about a singular-
ity. This involves taking a sequence of points and times approaching the
singularity, dilating in space and time, and translating in time to obtain a
sequence of solutions. Depending on the type of singularity, slightly different
methods must be employed to find suitable sequences of points and times
so that one obtains a singularity model which can yield information about
the geometry of the original solution near the singularity just prior to its
In Chapter 9, we consider Type I singularities, those where the curvature
is bounded proportionally to the inverse of the time remaining until the sin-
gularity time. This is the parabolically natural rate of singularity formation
for the flow. In this case, one sees the R x 5
cylinder (or its quotients) as
singularity models. The results in this chapter provide additional rigor to
the intuition behind neckpinches.
In the two appendices, we provide elementary background for tensor
calculus and some comparison geometry.
At present (December 2003) there has been sustained ex-
citement in the mathematical community over Perelman's recent ground-
breaking progress on Hamilton's Ricci flow program intended to resolve
Thurston's Geometrization Conjecture. Perelman's results allow one to view
some of the material in Chapters 8 and 9 in a new light. We have retained
that material in this volume for its independent interest, and hope to present
Perelman's work in a subsequent volume.
A guide for the hurried reader
The reader wishing to develop a nontechnical appreciation of the Ricci
flow program for 3-manifolds as efficiently as possible is advised to follow
the fast track outlined below.
In Chapter 1, read Section 1 for a brief introduction to the Geometriza-
In Chapter 2, the most important examples are the cigar, the neckpinch
and the degenerate neckpinch. Read the discussion of the cigar soliton in
Section 2. Read the statements of the main results on neckpinches derived
in Section 5. And read the heuristic discussion of degenerate neckpinches in
In Chapter 3, review the variation formulas derived in Section 1.
In Chapter 4, read the proof of the first scalar maximum principle (The-
orem 4.2) and at least the statements of the maximum principles that follow.
In Chapter 5, review the entropy estimates derived in Section 8 and the
differential Harnack estimates derived in Section 10. The entropy estimates