Geometrisation of 3-manifolds:
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Zürich
European Mathematical Society
[2010]
|
| Series: | EMS Tracts in mathematics
13 |
| Subjects: | |
| Links: | 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 |
| Physical Description: | X, 237 Seiten Illustrationen, Diagramme |
| ISBN: | 9783037190821 |
Staff View
MARC
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| 264 | 1 | |a Zürich |b European Mathematical Society |c [2010] | |
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Record in the Search Index
| _version_ | 1819375938042855424 |
|---|---|
| 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 | 2024-12-20T14:40:32Z |
| institution | BVB |
| isbn | 9783037190821 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020635869 |
| oclc_num | 699638214 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-188 DE-19 DE-BY-UBM DE-83 DE-11 |
| owner_facet | DE-355 DE-BY-UBR DE-188 DE-19 DE-BY-UBM DE-83 DE-11 |
| physical | X, 237 Seiten Illustrationen, Diagramme |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | European Mathematical Society |
| record_format | marc |
| series | EMS Tracts in mathematics |
| series2 | EMS Tracts in mathematics |
| spellingShingle | Bessières, Laurent Besson, Gérard 1955- Maillot, Sylvain Boileau, Michel Porti, Joan 1967- Geometrisation of 3-manifolds EMS Tracts in mathematics Kohomologietheorie (DE-588)4164610-1 gnd L-Funktion (DE-588)4137026-0 gnd Poincaré-Vermutung (DE-588)4517256-0 gnd Ricci-Fluss (DE-588)7531847-7 gnd Kompakte Mannigfaltigkeit (DE-588)4164848-1 gnd |
| subject_GND | (DE-588)4164610-1 (DE-588)4137026-0 (DE-588)4517256-0 (DE-588)7531847-7 (DE-588)4164848-1 |
| 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 | Kohomologietheorie (DE-588)4164610-1 gnd L-Funktion (DE-588)4137026-0 gnd Poincaré-Vermutung (DE-588)4517256-0 gnd Ricci-Fluss (DE-588)7531847-7 gnd Kompakte Mannigfaltigkeit (DE-588)4164848-1 gnd |
| topic_facet | Kohomologietheorie L-Funktion Poincaré-Vermutung Ricci-Fluss Kompakte Mannigfaltigkeit |
| 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 |
| volume_link | (DE-604)BV022480257 |
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