V l l l
model is a solution of the flow that arises as the limit of a sequence of dila-
tions of an original solution approaching a singularity.) Since the underlying
manifolds of such limit solutions are topologically simple, a detailed analy-
sis of the singularities which arise in dimension three is therefore possible.
For example, showing that certain singularity models are shrinking cylin-
ders is one of the cornerstones for enabling geometric-topological surgeries
to be performed on singular solutions to the Ricci flow on 3-manifolds. The
Li-Yau-Hamilton-type differential Harnack quantity is another innovation;
this yields an a priori estimate for a certain expression involving the cur-
vature and its first and second derivatives in space. This estimate allows
one to compare a solution at different points and times. In particular, it
shows that in the presence of a nonnegative curvature operator, the scalar
curvature does not decrease too fast. Another consequence of Hamilton's
differential Harnack estimate is that slowly forming singularities (those in
which the curvature of the original solution grows more quickly than the
parabolically natural rate) lead to singularity models that are stationary
The convergence theory of Cheeger and Gromov has had fundamental
consequences in Riemannian geometry. In the setting of the Ricci flow,
Gromov's compactness theorem may be improved to obtain C°° convergence
of a sequence of solutions to a smooth limit solution. For singular solutions,
Perelman's recent No Local Collapsing Theorem allows one to dilate about
sequences of points and times approaching the singularity time in such a way
that one can obtain a limit solution that exists infinitely far back in time.
The analysis of such limit solutions is important in Hamilton's singularity
A guide for the reader
The reader of this book is assumed to have a basic knowledge of Rie-
mannian geometry. Familiarity with algebraic topology and with nonlinear
second-order partial differential equations would also be helpful but is not
strictly necessary.
In Chapters 1 and 2, we begin the study of the Ricci flow by considering
special solutions which exhibit typical properties of the Ricci flow and which
guide our intuition. In the case of an initial homogeneous metric, the solution
remains homogenous so that its analysis reduces to the study of a system of
ODE. We study examples of such solutions in Chapter 1. Solutions which
exist on long time intervals such as those which exist since time oo or until
time +oo are very special and can appear as singularity models. In Chapter
2, we both present such solutions explicitly and provide some intuition for
how they arise. An important example of a stationary solution is the so-
called "cigar soliton" on
intuitive solutions include the neckpinch and
degenerate neckpinch. In Chapter 2, we discuss degenerate neckpinches
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