Inspired by Grigory Perelman's three preprints, although not slavishly devoted to Perelman's organization, Morgan and Tian provide full details of a complete proof of geometric arguments (particularly the Ricci flow as introduced and studied by Hamilton) that establish the Poincaré Conjecture in the affirmative. In this comprehensive and well-illustrated monograph they describe the background of Riemann geometry and Ricci flow, including manifolds of non-negative curvature, the maximum principle and convergence results for Ricci flow, Perelman's length function and its applications, including a comparative geometry approach to the Ricci flow, complete Ricci flows of bounded curvature, results and the standard solution, Ricci flow with surgery, and completion of the proof of the conjecture, including finite-time extinction. An appendix describes 3-manifolds covered by canonical neighborhoods. Annotation ©2007 Book News, Inc., Portland, OR (booknews.com)