Comments:

541 pages, 2006

Abstract:

The first complete proof of the important Poincar\'e 1904 conjecture is presented and explained in full detail based on the concept of Ricci flow, in part following William Thurston's approach in his Princeton lectures. The conjecture proposed the characterization of the 3-sphere, one of the fundamental questions in topology. According to the book authors: "The last part follows Perelman's third preprint to prove that when the initial Riemannian 3--manifold has a finite fundamental group, the Ricci flow with surgery becomes extinct after a finite time. The proofs of Poincar\'e Conjecture and the closely related 3-dimensional spherical space-form conjecture are then immediate." On the other hand, as the authors state: "the existence of Ricci flow with surgery has applications to 3-manifolds far beyond the Poincar\'e Conjecture. It forms the heart of the proof via Ricci flow of Thurston's Geometrization Conjecture",... that is not treated in this book but is planned for a follow up textbook: "Thurston's Geometrization Conjecture, which classifies all compact 3-manifolds, will be the subject of a follow-up article."

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http://www.claymath.org/publications/ricciflow/ AMS/CMS

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