Ricci Flow and the Poincare Conjecture
American Mathematical Society (Verlag)
978-1-4704-7316-7 (ISBN)
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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.
The existence of Ricci flow with surgery has application to 3-manifolds far beyond the Poincare Conjecture. It forms the heart of the proof via Ricci flow of Thurston's Geometrization Conjecture. Thurston's Geometrization Conjecture, which classifies all compact 3-manifolds, will be the subject of a follow-up article.
The organization of the material in this book differs from that given by Perelman. From the beginning the authors present all analytic and geometric arguments in the context of Ricci flow with surgery. In addition, the fourth part is a much-expanded version of Perelman's third preprint; it gives the first complete and detailed proof of the finite-time extinction theorem. 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. Clay Mathematics Institute Monograph Series The Clay Mathematics Institute Monograph Series publishes selected expositions of recent developments, both in emerging areas and in older subjects transformed by new insights or unifying ideas.
John Morgan, Columbia University, New York, NY. Gang Tian, Princeton University, NJ, and Peking University, Beijing, China.
Background from Riemannian geometry and Ricci flow: Preliminaries from Riemannian geometry
Manifolds of non-negative curvature
Basics of Ricci flow
The maximum principle
Convergence results for Ricci flow
Perelman's length function and its applications: A comparison geometry approach to the Ricci flow
Complete Ricci flows of bounded curvature
Non-collapsed results
$/kappa$-non-collapsed ancient solutions
Bounded curvature at bounded distance
Geometric limits of generalized Ricci flows
The standard solution
Ricci flow with surgery: Surgery on a $/delta$-neck
Ricci flow with surgery: The definition
Controlled Ricci flows with surgery
Proof of non-collapsing
Completion of the proof of Theorem 15.9
Completion of the proof of the Poincare conjecture: Finite-time extinction
Completion of the proof of Proposition 18.24
3-manifolds covered by canonical neighborhoods
Bibliography
Index
| Erscheinungsdatum | 23.09.2025 |
|---|---|
| Reihe/Serie | Clay Mathematics Monographs |
| Verlagsort | Providence |
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Geometrie / Topologie |
| ISBN-10 | 1-4704-7316-X / 147047316X |
| ISBN-13 | 978-1-4704-7316-7 / 9781470473167 |
| Zustand | Neuware |
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