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Geometrisation of 3-Manifolds

Laurent Bessières Université Joseph Fourier, Grenoble, France
Gérard Besson Université Joseph Fourier, Grenoble, France
Michel Boileau Université Paul Sabatier, Toulouse, France
Sylvain Maillot Université Montpellier II, Montpellier, France
Joan Porti Universitat Autonoma de Barcelona, Spain
A publication of European Mathematical Society
Available Formats:
Hardcover ISBN: 978-3-03719-082-1
Product Code: EMSTM/13
List Price: $64.00 AMS Member Price:$51.20
Please note AMS points can not be used for this product
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Geometrisation of 3-Manifolds
Laurent Bessières Université Joseph Fourier, Grenoble, France
Gérard Besson Université Joseph Fourier, Grenoble, France
Michel Boileau Université Paul Sabatier, Toulouse, France
Sylvain Maillot Université Montpellier II, Montpellier, France
Joan Porti Universitat Autonoma de Barcelona, Spain
A publication of European Mathematical Society
Available Formats:
 Hardcover ISBN: 978-3-03719-082-1 Product Code: EMSTM/13
 List Price: $64.00 AMS Member Price:$51.20
Please note AMS points can not be used for this product
• Book Details

EMS Tracts in Mathematics
Volume: 132010; 247 pp
MSC: Primary 57; 53;

The geometrisation conjecture was proposed by William Thurston in the mid 1970s in order to classify compact $3$-manifolds by means of a canonical decomposition along essential, embedded surfaces into pieces that possess geometric structures. It contains the famous Poincaré Conjecture as a special case.

In 2002 Grigory Perelman announced a proof of the geometrisation conjecture based on Richard Hamilton's Ricci flow approach and presented it in a series of three celebrated arXiv preprints. Since then there has been an ongoing effort to understand Perelman's work by giving more detailed and accessible presentations of his ideas or alternative arguments for various parts of the proof.

This book is a contribution to this endeavor. Its two main innovations are first a simplified version of Perelman's Ricci flow with surgery, which is called Ricci flow with bubbling-off, and secondly a completely different and original approach to the last step of the proof. In addition, special effort has been made to simplify and streamline the overall structure of the argument and make the various parts independent of one another.

A complete proof of the geometrisation conjecture is given, modulo pre-Perelman results on Ricci flow, Perelman's results on the $\mathcal L$-functional and $\kappa$-solutions, as well as the Colding–Minicozzi extinction paper. The book can be read by anyone already familiar with these results or willing to accept them as black boxes. The structure of the proof is presented in a lengthy introduction which does not require knowledge of geometric analysis. The bulk of the proof is the existence theorem for Ricci flow with bubbling-off, which is treated in parts I and II. Part III deals with the long-time behaviors of Ricci flow with bubbling-off. Part IV finishes the proof of the geometrisation conjecture.

Graduate students and research mathematicians interested in the geometrisation conjecture.

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Volume: 132010; 247 pp
MSC: Primary 57; 53;

The geometrisation conjecture was proposed by William Thurston in the mid 1970s in order to classify compact $3$-manifolds by means of a canonical decomposition along essential, embedded surfaces into pieces that possess geometric structures. It contains the famous Poincaré Conjecture as a special case.

In 2002 Grigory Perelman announced a proof of the geometrisation conjecture based on Richard Hamilton's Ricci flow approach and presented it in a series of three celebrated arXiv preprints. Since then there has been an ongoing effort to understand Perelman's work by giving more detailed and accessible presentations of his ideas or alternative arguments for various parts of the proof.

This book is a contribution to this endeavor. Its two main innovations are first a simplified version of Perelman's Ricci flow with surgery, which is called Ricci flow with bubbling-off, and secondly a completely different and original approach to the last step of the proof. In addition, special effort has been made to simplify and streamline the overall structure of the argument and make the various parts independent of one another.

A complete proof of the geometrisation conjecture is given, modulo pre-Perelman results on Ricci flow, Perelman's results on the $\mathcal L$-functional and $\kappa$-solutions, as well as the Colding–Minicozzi extinction paper. The book can be read by anyone already familiar with these results or willing to accept them as black boxes. The structure of the proof is presented in a lengthy introduction which does not require knowledge of geometric analysis. The bulk of the proof is the existence theorem for Ricci flow with bubbling-off, which is treated in parts I and II. Part III deals with the long-time behaviors of Ricci flow with bubbling-off. Part IV finishes the proof of the geometrisation conjecture.