# Ricci Flow and the Poincaré Conjecture

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*John Morgan; Gang Tian*

A co-publication of the AMS and Clay Mathematics Institute

For over 100 years the Poincaré
Conjecture, which proposes a topological characterization of the
3-sphere, has been the central question in topology. Since its
formulation, it has been repeatedly attacked, without success, using
various topological methods. Its importance and difficulty were
highlighted when it was chosen as one of the Clay Mathematics
Institute's seven Millennium Prize Problems. In 2002 and 2003 Grigory
Perelman posted three preprints showing how to use geometric
arguments, in particular the Ricci flow as introduced and studied by
Hamilton, to establish the Poincaré Conjecture in the
affirmative.

This book provides full details of a complete proof of the
Poincaré Conjecture following Perelman's three preprints. After
a lengthy introduction that outlines the entire argument, the book is
divided into four parts. The first part reviews necessary results from
Riemannian geometry and Ricci flow, including much of Hamilton's
work. The second part starts with Perelman's length function, which is
used to establish crucial non-collapsing theorems. Then it discusses
the classification of non-collapsed, ancient solutions to the Ricci
flow equation. The third part concerns the existence of Ricci flow
with surgery for all positive time and an analysis of the topological
and geometric changes introduced by surgery. The last part follows
Perelman's third preprint to prove that when the initial Riemannian
3-manifold has finite fundamental group, Ricci flow with surgery
becomes extinct after finite time. The proofs of the Poincaré
Conjecture and the closely related 3-dimensional spherical space-form
conjecture are then immediate.

The existence of Ricci flow with surgery has application to
3-manifolds far beyond the Poincaré Conjecture. It forms the
heart of the proof via Ricci flow of Thurston's Geometrization
Conjecture. Thurston's Geometrization Conjecture, which classifies all
compact 3-manifolds, will be the subject of a follow-up article.

The organization of the material in this book differs from that
given by Perelman. From the beginning the authors present all analytic
and geometric arguments in the context of Ricci flow with surgery. In
addition, the fourth part is a much-expanded version of Perelman's
third preprint; it gives the first complete and detailed proof of the
finite-time extinction theorem.

With the large amount of background material that is presented and
the detailed versions of the central arguments, this book is suitable
for all mathematicians from advanced graduate students to specialists
in geometry and topology.

The Clay Mathematics Institute Monograph Series publishes selected
expositions of recent developments, both in emerging areas and in
older subjects transformed by new insights or unifying
ideas.

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

#### Readership

Graduate students and research mathematicians interested in geometry and topology.

#### Reviews & Endorsements

The comprehensive and carefully detailed nature of the text makes this book an invaluable resource for any mathematician who wants to understand the technical nuts and bolts of the proof, while the introductory chapter provides an excellent conceptual overview of the entire argument.

-- Mathematical Reviews