Preface This book and its planned sequel(s) are intended to compose an introduc- tion to the Ricci flow in general and in particular to the program originated by Hamilton to apply the Ricci flow to approach Thurston's Geometrization Conjecture. The Ricci flow is the geometric evolution equation in which one starts with a smooth Riemannian manifold {Mn,go) and evolves its metric by the equation | , = -2Rc, where Re denotes the Ricci tensor of the metric g. The Ricci flow was introduced in Hamilton's seminal 1982 paper, "Three-manifolds with pos- itive Ricci curvature". In this paper, closed 3-manifolds of positive Ricci curvature are topologically classified as spherical space forms. Until that time, most results relating the curvature of a 3-manifold to its topology in- volved the influence of curvature on the fundamental group. For example, Gromov-Lawson and Schoen-Yau classified 3-manifolds with positive scalar curvature essentially up to homotopy. Among the many results relating cur- vature and topology that hold in arbitrary dimensions are Myers' Theorem and the Cartan-Hadamard Theorem. A large number of innovations that originated in Hamilton's 1982 and subsequent papers have had a profound influence on modern geometric anal- ysis. Here we mention just a few. Hamilton's introduction of a nonlinear heat-type equation for metrics, the Ricci flow, was motivated by the 1964 harmonic heat flow introduced by Eells and Sampson. This led to the re- newed study by Huisken, Ecker, and many others of the mean curvature flow originally studied by Brakke in 1977. One of the techniques that has dominated Hamilton's work is the use of the maximum principle, both for functions and for tensors. This technique has been applied to control vari- ous geometric quantities associated to the metric under the Ricci flow. For example, the so-called pinching estimate for 3-manifolds with positive cur- vature shows that the eigenvalues of the Ricci tensor approach each other as the curvature becomes large. (This result is useful because the Ricci flow shrinks manifolds with positive Ricci curvature, which tends to make the curvature larger.) Another curvature estimate, due independently to Ivey and Hamilton, shows that the singularity models that form in dimen- sion three necessarily have nonnegative sectional curvature. (A singularity V l l
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