Geometrisation of 3-manifolds:
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| Main Authors: | , , , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Zürich
European Mathematical Society
[2010]
|
| Series: | EMS Tracts in mathematics
13 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 237 Seiten Illustrationen, Diagramme |
| ISBN: | 9783037190821 |
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MARC
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| 245 | 1 | 0 | |a Geometrisation of 3-manifolds |c Laurent Bessières, Gérard Besson, Sylvain Maillot, Michel Boileau, Joan Porti |
| 264 | 1 | |a Zürich |b European Mathematical Society |c [2010] | |
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Record in the Search Index
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| adam_text |
Contents
Preface
v
1
The
Géométrisation 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
géométrisation
. 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
géométrisation
. 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
ε
-necks.
35
4
«r-noncollapsing, canonical geometry and pinching
37
4.1
AT-noncollapsing
. 37
4.2
ic-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
viii Contents
4.4
Curvature pinched toward positive
. 43
5
Ricci
flow with (r,
5,
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,
8,
/c)-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
Π
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-noncoIIapsing and the proof of Proposition
С
108
10.1
Preliminaries
. 109
10.1.1
Basic facts on /c-noncollapsing
. 109
10.1.2
Perelman's «Z-length
.
Ill
10.2
Proof of Theorem
10.0.3. 112
10.3
/r-noneollapsing of
Ricci
flow with bubbling-off: proof of
Proposition
С
. 116
10.3.1
The case p0
<
ř
. 121
10.3.2
The case p0
ž
ř .
123
10.4
к
-noncollapsing at bounded distance of the thick part
. 125
Contents ix
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-manifołds
207
χ
Contents
Appendix
A 3-manifold topology
209
A.I 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.I Comparison and compactness theorems
. 213
B.2 Manifolds with
nonnegative
curvature
. 214
AppendixC
Ricci
flow
217
C.I 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
Б
A sufficient condition for
hyperbolicky
222
Bibliography
225
Index
235 |
| any_adam_object | 1 |
| author | Bessières, Laurent Besson, Gérard 1955- Maillot, Sylvain Boileau, Michel Porti, Joan 1967- |
| author_GND | (DE-588)1022464868 (DE-588)125464587X (DE-588)133798003X (DE-588)173132189 |
| author_facet | Bessières, Laurent Besson, Gérard 1955- Maillot, Sylvain Boileau, Michel Porti, Joan 1967- |
| author_role | aut aut aut aut aut |
| author_sort | Bessières, Laurent |
| author_variant | l b lb g b gb s m sm m b mb j p jp |
| building | Verbundindex |
| bvnumber | BV036717935 |
| classification_rvk | SK 780 |
| ctrlnum | (OCoLC)699638214 (DE-599)BVBBV036717935 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV036717935 |
| illustrated | Illustrated |
| indexdate | 2024-09-26T12:01:26Z |
| institution | BVB |
| isbn | 9783037190821 |
| language | English |
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| physical | X, 237 Seiten Illustrationen, Diagramme |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | European Mathematical Society |
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| series | EMS Tracts in mathematics |
| series2 | EMS Tracts in mathematics |
| spelling | Bessières, Laurent Verfasser (DE-588)1022464868 aut Geometrisation of 3-manifolds Laurent Bessières, Gérard Besson, Sylvain Maillot, Michel Boileau, Joan Porti Zürich European Mathematical Society [2010] © 2010 X, 237 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier EMS Tracts in mathematics 13 Kompakte Mannigfaltigkeit (DE-588)4164848-1 gnd rswk-swf Ricci-Fluss (DE-588)7531847-7 gnd rswk-swf Poincaré-Vermutung (DE-588)4517256-0 gnd rswk-swf Kohomologietheorie (DE-588)4164610-1 gnd rswk-swf L-Funktion (DE-588)4137026-0 gnd rswk-swf Kohomologietheorie (DE-588)4164610-1 s L-Funktion (DE-588)4137026-0 s DE-604 Kompakte Mannigfaltigkeit (DE-588)4164848-1 s Poincaré-Vermutung (DE-588)4517256-0 s Ricci-Fluss (DE-588)7531847-7 s Besson, Gérard 1955- Verfasser (DE-588)125464587X aut Maillot, Sylvain Verfasser aut Boileau, Michel Verfasser (DE-588)133798003X aut Porti, Joan 1967- Verfasser (DE-588)173132189 aut Erscheint auch als Online-Ausgabe 978-3-03719-582-6 (DE-604)BV036739651 EMS Tracts in mathematics 13 (DE-604)BV022480257 13 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020635869&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Geometrisation of 3-manifolds EMS Tracts in mathematics Kompakte Mannigfaltigkeit (DE-588)4164848-1 gnd Ricci-Fluss (DE-588)7531847-7 gnd Poincaré-Vermutung (DE-588)4517256-0 gnd Kohomologietheorie (DE-588)4164610-1 gnd L-Funktion (DE-588)4137026-0 gnd Bessières, Laurent Besson, Gérard 1955- Maillot, Sylvain Boileau, Michel Porti, Joan 1967- |
| subject_GND | (DE-588)4164848-1 (DE-588)7531847-7 (DE-588)4517256-0 (DE-588)4164610-1 (DE-588)4137026-0 |
| title | Geometrisation of 3-manifolds |
| title_auth | Geometrisation of 3-manifolds |
| title_exact_search | Geometrisation of 3-manifolds |
| title_full | Geometrisation of 3-manifolds Laurent Bessières, Gérard Besson, Sylvain Maillot, Michel Boileau, Joan Porti |
| title_fullStr | Geometrisation of 3-manifolds Laurent Bessières, Gérard Besson, Sylvain Maillot, Michel Boileau, Joan Porti |
| title_full_unstemmed | Geometrisation of 3-manifolds Laurent Bessières, Gérard Besson, Sylvain Maillot, Michel Boileau, Joan Porti |
| title_short | Geometrisation of 3-manifolds |
| title_sort | geometrisation of 3 manifolds |
| topic | Kompakte Mannigfaltigkeit (DE-588)4164848-1 gnd Ricci-Fluss (DE-588)7531847-7 gnd Poincaré-Vermutung (DE-588)4517256-0 gnd Kohomologietheorie (DE-588)4164610-1 gnd L-Funktion (DE-588)4137026-0 gnd |
| topic_facet | Kompakte Mannigfaltigkeit Ricci-Fluss Poincaré-Vermutung Kohomologietheorie L-Funktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020635869&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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