Description
Book SynopsisFor 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 attacked, without success, using various topological methods. Its importance and difficulty were highlighted when it was chosen as one of the Clay Mathematics Institute's seven Millennium Prize Problems. In 2002 and 2003 Grigory Perelman posted three preprints showing how to use geometric arguments, in particular the Ricci flow as introduced and studied by Hamilton, to establish the Poincare Conjecture in the affirmative. 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 le
Trade ReviewThe comprehensive and carefully detailed nature of the text makes this book an invaluable resource for any mathematician who wants to understand the technical nuts and bolts of the proof, while the introductory chapter provides an excellent conceptual overview of the entire argument." —
Mathematical Reviews Table of Contents
- 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
