Preface ......................................................... v
1 The Geometrisation Conjecture ................................ 1
1.1 Introduction ............................................ 1
1.2 Ricci flow and elliptisation ............................ 5
1.2.1 Ricci flow ....................................... 5
1.2.2 Ricci flow with bubbling-off ..................... 6
1.2.3 Application to elliptisation ..................... 9
1.3 3-manifolds with infinite fundamental group ............ 11
1.3.1 Long-time behaviour of the Ricci flow with
bubbling-off .................................... 11
1.3.2 Hyperbolisation ................................. 13
1.4 Some consequences of geometrisation .................... 15
1.4.1 The homeomorphism problem ....................... 15
1.4.2 Fundamental group ............................... 17
1.5 Final remarks .......................................... 21
1.5.1 Comparison with Perelman's original arguments ... 21
1.5.2 Beyond geometrisation ........................... 21
Part I Ricci flow with bubbling-off: definitions and
statements ..................................................... 23
2 Basic definitions ........................................... 25
2.1 Riemannian geometry conventions ........................ 25
2.2 Evolving metrics and Ricci flow with bubbling-off ...... 26
3 Piecing together necks and caps ............................. 31
3.1 Necks, caps and tubes .................................. 31
3.1.1 Necks ........................................... 31
3.1.2 Caps and tubes .................................. 31
3.2 Gluing results ......................................... 32
3.3 More results on e-necks ................................ 35
4 k-noncollapsing, canonical geometry and pinching ............ 37
4.1 k-noncollapsing ........................................ 37
4.2 k-solutions ............................................ 38
4.2.1 Definition and main results ..................... 38
4.2.2 Canonical neighbourhoods ........................ 39
4.3 The standard solution I ................................ 40
4.3.1 Definition and main results ..................... 40
4.3.2 Neck strengthening .............................. 41
4.4 Curvature pinched toward positive ....................... 43
5 Ricci flow with (r, δ, k)-bubbling-off ...................... 46
5.1 Let the constants be fixed ............................. 46
5.2 Metric surgery and cutoff parameters ................... 47
5.3 Finite-time existence theorem for Ricci flow with (r,
δ, k)-bubbling-off .................................... 50
5.3.1 The statements .................................. 50
5.3.2 Proof of the finite-time existence theorem,
assuming Propositions А, В, С ................... 51
5.4 Long-time existence of Ricci flow with bubbling-off .... 54
Part II Ricci flow with bubbling-off: existence ............... 57
6 Choosing cutoff parameters .................................. 59
6.1 Bounded curvature at bounded distance .................. 59
6.1.1 Preliminaries ................................... 60
6.1.2 Proof of Curvature-Distance Theorem 6.1.1 ....... 62
6.2 Existence of cutoff parameters ......................... 70
7 Metric surgery and the proof of Proposition A ............... 75
7.1 The standard solution II ............................... 75
7.2 Proof of the metric surgery theorem .................... 79
7.3 Proof of Proposition A ................................. 86
8 Persistence ................................................. 88
8.1 Introduction ........................................... 88
8.2 Persistence of a model ................................. 90
8.3 Application: persistence of almost standard caps ....... 95
9 Canonical neighbourhoods and the proof of Proposition В ..... 97
9.1 Warming up ............................................. 97
9.2 The proof .............................................. 98
10 k-noncollapsing and the proof of Proposition С ............. 108
10.1 Preliminaries ......................................... 109
10.1.1 Basic facts on k-noncollapsing ................. 109
10.1.2 Perelman's  -length ............................ 111
10.2 Proof of Theorem 10.0.3 ............................... 112
10.3 k-noncollapsing of Ricci flow with bubbling-off:
proof of Proposition С ................................ 116
10.3.1 The case ρ0 <  ................................ 121
10.3.2 The case ρ0 ≥ ............................... 123
10.4 k-noncollapsing at bounded distance of the thick
part .................................................. 125
10.4.1 A formal computation ........................... 126
10.4.2 Justification of the formal computations ....... 127
Part III Long-time behaviour of Ricci flow with
bubbling-off .................................................. 131
11 The thin-thick decomposition theorem ....................... 133
11.1 Introduction: main statements ......................... 133
11.2 Proof of the thin-thick decomposition theorem ......... 136
11.2.1 Rescaled volume is bounded and limits are
hyperbolic ..................................... 136
11.2.2 Hyperbolic limits exist: proof of part (ii) .... 139
11.2.3 Locally controlled curvature: proof of part
(iii) .......................................... 143
12 Refined estimates for long-time behaviour .................. 146
12.1 Spatial extension of local estimates: proof of
Theorem 11.1.3 ........................................ 146
12.1.1 Canonical neighbourhoods: proof of part (b) .... 146
12.1.2 Curvature-distance estimates: proof of part
(c) ............................................ 153
12.2 Curvature estimates in the thick part: proof of
Theorem 11.1.6 ........................................ 156
Part IV Weak collapsing and hyperbolisation ................... 177
13 Collapsing, simplicial volume and strategy of proof ........ 179
13.1 Collapsing and weak collapsing ........................ 179
13.2 Simplicial volume ..................................... 181
13.2.1 Definition and first examples .................. 182
13.2.2 Simplicial volume and geometric
decompositions ................................. 182
13.2.3 Simplicial volume and collapsing ............... 183
13.3 Sketch of proof of Theorem 13.1.3 ..................... 184
13.3.1 The collapsing case ............................ 184
13.3.2 The general case ............................... 185
13.4 Comments .............................................. 185
14 Proof of the weak collapsing theorem ....................... 188
14.1 Structure of the thick part ........................... 188
14.2 Local structure of the thin part ...................... 190
14.3 Constructions of coverings ............................ 195
14.3.1 Embedding thick pieces in solid tori ........... 195
14.3.2 Existence of a homotopically nontrivial open
set ............................................ 195
14.3.3 End of the proof: covering by virtually
abelian subsets ................................ 201
15 A rough classification of 3-manifolds ...................... 207
Appendix A 3-manifold topology ............................... 209
A.1 General notation ...................................... 209
A.2 Alexander's theorem and consequences .................. 209
A.3 Submanifolds with compressible boundary ............... 210
A.4 Covering 3-manifolds by abelian subsets ............... 211
Appendix В Comparison geometry ............................... 213
B.l Comparison and compactness theorems ................... 213
B.2 Manifolds with nonnegative curvature .................. 214
Appendix С Ricci flow ......................................... 217
C.l Existence and basic properties ........................ 217
C.2 Consequences of the maximum principle ................. 217
C.3 Compactness ........................................... 218
C.4 Harnack inequalities for the Ricci Flow ............... 219
C.5 Ricci Flow on cones ................................... 220
Appendix D Alexandrov spaces .................................. 221
Appendix E A sufficient condition for hyperbolicity ........... 222
Bibliography .................................................. 225
Index ......................................................... 235
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