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
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