# The Geometrization Conjecture

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*John Morgan; Gang Tian*

A co-publication of the AMS and Clay Mathematics Institute

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).

#### Readership

Graduate students and research mathematicians interested in topology and geometry.