The Geometrization Conjecture

Front Cover
American Mathematical Soc., May 21, 2014 - Mathematics - 291 pages

This book gives a complete proof of the geometrization conjecture, which describes all compact 3-manifolds in terms of geometric pieces, i.e., 3-manifolds with locally homogeneous metrics of finite volume. The method is to understand the limits as time goes to infinity of Ricci flow with surgery. The first half of the book is devoted to showing that these limits divide naturally along incompressible tori into pieces on which the metric is converging smoothly to hyperbolic metrics and pieces that are locally more and more volume collapsed. The second half of the book is devoted to showing that the latter pieces are themselves geometric. This is established by showing that the Gromov-Hausdorff limits of sequences of more and more locally volume collapsed 3-manifolds are Alexandrov spaces of dimension at most 2 and then classifying these Alexandrov spaces.

In the course of proving the geometrization conjecture, the authors provide an overview of the main results about Ricci flows with surgery on 3-dimensional manifolds, introducing the reader to this difficult material. The book also includes an elementary introduction to Gromov-Hausdorff limits and to the basics of the theory of Alexandrov spaces. In addition, a complete picture of the local structure of Alexandrov surfaces is developed. All of these important topics are of independent interest.

Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).

 

 

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Contents

Introduction
1
GEOMETRIC AND ANALYTIC RESULTS
11
Limits as t
25
Local results valid for large time
41
Proofs of the three propositions
66
Part 2
95
The collapsing theorem
101
Overview of the rest of the argument
107
3dimensional analogues
163
LOCALLY VOLUME COLLAPSED 3MANIFOLDS
181
13
205
14
226
The equivariant case
245
Bibliography
281
21
282
93
289

Basics of Alexandrov spaces
117
2dimensional Alexandrov spaces
141

Common terms and phrases

About the author (2014)

John Morgan, Simons Center for Geometry and Physics, Stony Brook University, NY.

Gang Tian, Princeton University, NJ, and Peking University, Beijing, China.

Bibliographic information