The Geometrization Conjecture

John Morgan, Gang Tian (Autoren)

Buch | Hardcover

291 Seiten

2014
American Mathematical Society (Verlag)
978-0-8218-5201-9 (ISBN)

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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.

John Morgan, Simons Center for Geometry and Physics, Stony Brook University, NY. Gang Tian, Princeton University, NJ, and Peking University, Beijing, China.

Introduction
Geometric and analytic results for Ricci flow with surgery
Ricci flow with surger
Limits as t??
Local results valid for large time
Proofs of the three propositions
Locally volume collapsed 3-manifolds
Introduction to part II
The collapsing theorem
Overview of the rest of the argument
Basics of Gromov-Hausdorff convergence
Basics of Alexandrov spaces
2-dimensional Alexandrov spaces
3-dimensional analogues
The global result
The equivariant case
The equivariant case
Bibliography
Glossary of symbols
Index

Erscheint lt. Verlag	22.9.2014
Reihe/Serie	Clay Mathematics Monographs
Verlagsort	Providence
Sprache	englisch
Maße	178 x 254 mm
Gewicht	692 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
ISBN-10	0-8218-5201-9 / 0821852019
ISBN-13	978-0-8218-5201-9 / 9780821852019
Zustand	Neuware