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Ricci flow and the Poincaré conjecture

Author: John Morgan; Gang Tian
Publisher: Providence, RI : American Mathematical Society : Clay Mathematics Institute, ©2007.
Series: Clay mathematics monographs, v. 3.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
"This book provides full details of a complete proof of the Poincare Conjecture following Perelman's three preprints. After a lengthy introduction that outlines the entire argument, the book is divided into four parts. The first part reviews necessary results from Riemannian geometry and Ricci flow, including much of Hamilton's work. The second part starts with Perelman's length function, which is used to establish  Read more...
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Details

Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: John Morgan; Gang Tian
ISBN: 9780821843284 0821843281
OCLC Number: 145147251
Description: xlii, 521 pages : illustrations ; 27 cm.
Contents: pt. 1. Background from Riemannian geometry and Ricci flow --
Ch. 1. Preliminaries from Riemannian geometry --
Ch. 2. Manifolds of non-negative curvature --
Ch. 3. Basics of Ricci flow --
Ch. 4. The maximum principle --
Ch. 5. Convergence results for Ricci flow --
pt. 2. Perelman's length function and its applications --
Ch. 6. A comparison geometry approach to the Ricci flow --
Ch. 7. Complete Ricci flows of bounded curvature --
Ch. 8. Non-collapsed results --
Ch. 9. [kappa]-non-collapsed ancient solutions --
Ch. 10. Bounded curvature at bounded distance --
Ch. 11. Geometric limits of generalized Ricci flows --
Ch. 12. The standard solution --
pt. 3. Ricci flow with surgery --
Ch. 13. Surgery on a [delta]-neck --
Ch. 14. Ricci flow with surgery : the definition --
Ch. 15. Controlled Ricci flows with surgery --
Ch. 16. Proof of non-collapsing --
Ch. 17. Completion of the proof of Theorem 15.9 --
pt. 4. Completion of the proof of the Poincare conjecture --
Ch. 18. Finite-time extinction --
Ch. 19. Completion of the proof of proposition 18.24 --
App. 3-manifolds covered by canonical neighborhoods.
Series Title: Clay mathematics monographs, v. 3.
Responsibility: John Morgan, Gang Tian.

Abstract:

For over 100 years the Poincare Conjecture, which proposes a topological characterization of the 3-sphere, has been the central question in topology. Since its formulation, it has been repeatedly  Read more...

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   schema:reviewBody ""This book provides full details of a complete proof of the Poincare Conjecture following Perelman's three preprints. After a lengthy introduction that outlines the entire argument, the book is divided into four parts. The first part reviews necessary results from Riemannian geometry and Ricci flow, including much of Hamilton's work. The second part starts with Perelman's length function, which is used to establish crucial non-collapsing theorems. Then it discusses the classification of non-collapsed, ancient solutions to the Ricci flow equation. The third part concerns the existence of Ricci flow with surgery for all positive time and an analysis of the topological and geometric changes introduced by surgery. The last part follows Perelman's third preprint to prove that when the initial Riemannian 3-manifold has finite fundamental group, Ricci flow with surgery becomes extinct after finite time. The proofs of the Poincare Conjecture and the closely related 3-dimensional spherical space-form conjecture are then immediate." "With the large amount of background material that is presented and the detailed versions of the central arguments, this book is suitable for all mathematicians from advanced graduate students to specialists in geometry and topology."--Jacket." ;
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