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.
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