The geometrization conjecture
By: John Morgan, Gang Tian
Contributors: Tian, Gang, 1958- author
Series: Clay mathematics monographs -- volume 5
Publisher: American Mathematical Society; CMI, Clay Mathematics Institute
Published: 2014
Place of publication: Providence, Rhode Island
Language: eng
Number of pages: 291
ISBN-13: 9780821852019
Copyright date: 2014
Subjects: Global Riemannian geometry; Topological manifolds; Differential geometry -- Global differential geometry -- Methods of Riemannian geometry, including PDE methods; curvature restrictions; Differential geometry -- Global differential geometry -- Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces; Differential geometry -- Global differential geometry -- Homogeneous manifolds; Differential geometry -- Global differential geometry -- Geometric evolution equations (mean curvature flow, Ricci flow, etc.).; Differential geometry -- Global differential geometry -- Global surface theory (convex surfaces à la A. D. Aleksandrov); Manifolds and cell complexes -- Low-dimensional topology -- Group actions in low dimensions
Description: 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 --Provided by publisher.
Notes: Includes bibliographical references (pages 284-282) and index.
Table of contents: title: Introduction; title: Ricci flow with surgery; title: Limits as [t goes to infinity]; title: Local results valid for large time; title: Proofs of the three propositions; title: Introduction to part II; title: The collapsing theorem; title: Overview of the rest of the argument; title: Basics of Gromov-Hausdorff convergence; title: Basics of Alexandrov spaces; title: 2-dimensional Alexandrov spaces; title: 3-dimensional analogues; title: The global result; title: The equivariant case.