Geometrisation of 3-Manifolds (EMS Tracts in Mathematics)

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 endeavour. 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 L-functional and κ-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 behaviour of Ricci flow with bubbling-off. Part IV finishes the proof of the Geometrisation Conjecture. Preface......Page 5 Introduction......Page 11 Ricci flow and elliptisation......Page 14 Ricci flow......Page 15 Ricci flow with bubbling-off......Page 16 Application to elliptisation......Page 19 Long-time behaviour of the Ricci flow with bubbling-off......Page 21 Hyperbolisation......Page 23 The homeomorphism problem......Page 25 Fundamental group......Page 27 Final remarks......Page 30 Beyond geometrisation......Page 31 Part I Ricci flow with bubbling-off: definitions and statements......Page 33 Riemannian geometry conventions......Page 35 Evolving metrics and Ricci flow with bubbling-off......Page 36 Gluing results......Page 41 More results on -necks......Page 45 kappa-noncollapsing......Page 47 Definition and main results......Page 48 Canonical neighbourhoods......Page 49 Definition and main results......Page 50 Neck strengthening......Page 51 Curvature pinched toward positive......Page 52 Let the constants be fixed......Page 56 Metric surgery and cutoff parameters......Page 57 The statements......Page 60 Proof of the finite-time existence theorem, assuming Propositions A, B, C......Page 61 Long-time existence of Ricci flow with bubbling-off......Page 64 Part II Ricci flow with bubbling-off: existence......Page 67 Bounded curvature at bounded distance......Page 69 Preliminaries......Page 70 Proof of Curvature-Distance Theorem 6.1.1......Page 72 Existence of cutoff parameters......Page 80 The standard solution II......Page 85 Proof of the metric surgery theorem......Page 89 Proof of Proposition A......Page 96 Introduction......Page 98 Persistence of a model......Page 100 Application: persistence of almost standard caps......Page 105 Warming up......Page 107 The proof......Page 108 10 kappa-noncollapsing and the proof of Proposition C......Page 118 Basic facts on -noncollapsing......Page 119 Perelman's ==========L-length......Page 121 Proof of Theorem 10.0.3......Page 122 kappa-noncollapsing of Ricci flow with bubbling-off: proof of Proposition C......Page 126 The case _0