A GUIDE FOR THE READER ix

heuristically. We also rigorously construct neckpinch solutions under certain

symmetry assumptions.

The short-time existence theorem for the Ricci flow with an arbitrary

smooth initial metric is proved in Chapter 3. This basic result allows one to

use the Ricci flow as a practical tool. In particular, a number of smoothing

results in Riemannian geometry can be proved using the short-time existence

of the flow combined with the derivative estimates of Chapter 7. Since the

Ricci flow system of equations is only weakly parabolic, the short-time ex-

istence of the flow does not follow directly from standard parabolic theory.

Hamilton's original proof relied on the Nash-Moser inverse function theo-

rem. Shortly thereafter, DeTurck gave a simplified proof by showing that

the Ricci flow is equivalent to a strictly parabolic system.

In Chapter 4, the maximum principle for both functions and tensors is

presented. This provides the technical foundations for the curvature, gradi-

ent of scalar curvature, and Li-Yau-Hamilton differential Harnack estimates

proved later in the book.

In Chapter 5, we give a comprehensive treatment of the Ricci flow on

surfaces. In this lowest nontrivial dimension, many of the techniques used

in three and higher dimensions are exhibited. The concept of stationary

solutions is introduced here and used to motivate the construction of var-

ious quantities, including Li-Yau-Hamilton (LYH) differential Harnack es-

timates. The entropy estimate is applied to solutions on the 2-sphere. In

addition, derivative and injectivity radius estimates are first proved here. We

also discuss the isoperimetric estimate and combine it with the Gromov-type

compactness theorem for solutions of the Ricci flow to rule out the cigar soli-

ton as a singularity model for solutions on surfaces. In higher dimensions,

the cigar soliton is ruled out by Perelman's No Local Collapsing Theorem.

The original topological classification by Hamilton of closed 3-manifolds

with positive Ricci curvature is proved in Chapter 6. Via the maximum

principle for systems, the qualitative behavior of the curvature tensor may

be reduced in this case to the study of a system of three ODE in three un-

knowns. Using this method, we prove the pinching estimate for the curvature

mentioned above, which shows that the eigenvalues of the Ricci tensor are

approaching each other at points where the scalar curvature is becoming

large. This pinching estimate compares curvatures at the same point. Then

we estimate the gradient of the scalar curvature in order to compare the cur-

vatures at different points. The combination of these estimates shows that

the Ricci curvatures are tending to constant. Using this fact together with

estimates for the higher derivatives of curvature, we prove the convergence

of the volume-normalized Ricci flow to a spherical space form.

The derivative estimates of Chapter 7 show that assuming an initial

curvature bound allows one to bound all derivatives of the curvature for a

short time, with the estimate deteriorating as the time tends to zero. (This

deterioration of the estimate is necessary, because an initial bound on the

curvature alone does not imply simultaneous bounds on its derivatives.) The