Ricci flow and geometrization of 3-manifolds:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2010]
|
| Series: | University lecture series
Volume 53 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | ix, 150 Seiten Diagramme |
| ISBN: | 9780821849637 |
Staff View
MARC
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| 100 | 1 | |a Morgan, John W. |d 1946- |e Verfasser |0 (DE-588)129352446 |4 aut | |
| 245 | 1 | 0 | |a Ricci flow and geometrization of 3-manifolds |c John W. Morgan ; Frederick Tsz-Ho Fong |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2010] | |
| 264 | 4 | |c © 2010 | |
| 300 | |a ix, 150 Seiten |b Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a University lecture series |v Volume 53 | |
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| 700 | 1 | |a Fong, Frederick Tsz-Ho |d 1983- |e Verfasser |0 (DE-588)142438731 |4 aut | |
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Record in the Search Index
| _version_ | 1804142797417086976 |
|---|---|
| adam_text | Contents
Preface
ix
Part
1.
Overview
1
Lecture
1 3
Geometric manifolds
3
Thurston manifolds
4
The theorems
5
Lecture
2 7
Basics of Riemannian geometry
7
Basics of
Ricci
flow
8
Canonical Neighborhoods
10
Lecture
3 13
More on Canonical Neighborhoods
13
Surgery on
Ricci
flow
14
Topologica!
effects of surgery
16
Lecture
4 17
More structure (geometric and analytic) of Canonical Neighborhoods
17
Finite-time extinction
18
Lecture
5 21
Geometric limits
21
Hyperbolic limits
22
The thin part
23
Alexandrov spaces
23
Summary of Part
1 25
Part
2.
Non-collapsing Results for
Ricci
Flows
27
Lecture
6 29
Geometric limits in the context of
Ricci
flow
29
Sketch of proof of the convergence theorem
31
Lecture
7 33
Non-collapsing: the statement
33
The ¿-function and ¿-geodesies
34
Lecture
8 37
vi
CONTENTS
The ¿-exponential map
37
Jacobi fields and the differential of C-exp
38
Lecture
9 41
Harnack s inequality
41
Relation of H(X) to ¿-geodesies
42
Lecture
10 45
More derivative estimates for
С
45
Hessian inequality
47
Lecture
11 49
Monotonicity
49
Example of
Ж
50
Non-collapsing of reduced volume
51
Lecture
12 53
Non-collapsing
53
Completion of proof
53
Part
3.
«-solutions
57
Lecture
13 59
Curvature pinching in dimension
3 59
Shrinking solitons
59
Lecture
14 63
Study of the length functions in a «-solution
63
Extensions of the inequalities
64
Convergence as
τ
—>
oo
64
Lecture
15 67
Proof of the existence of an asymptotic gradient shrinking soliton
67
Enhanced gradient shrinking solitons
70
Lecture
16 73
Toponogov s splitting theorem
73
Classification of asymptotic gradient shrinking solitons
73
Lecture
17 77
Asymptotic volume ratio and asymptotic curvature
77
Asymptotic curvature of
a
«-solution
77
Asymptotic volume ratio for a «-solution
78
Lecture
18 81
Compactness of the space of «-solutions
81
Proof of the compactness theorem for «-solutions
82
Lecture
19 85
Review of compactness of
З
-dimensional
«-solutions
85
Qualitative properties of «-solutions
86
Geometry of 3-dimensional «-solutions
87
CONTENTS
vii
Part
4.
The Canonical Neighborhood Theorem
89
Lecture
20 91
Blow-up limits
91
Canonical neighborhood theorem
92
Lecture
21 97
Completion of the proof of the canonical neighborhood theorem
97
Step
2
of proof
97
Step
3
of proof
97
Lecture
22 101
Review of proof
101
Additive distance inequality
101
Part
5.
Ricci
Flow with Surgery
105
Lecture
23 107
What happens at Tmax?
107
є
-horns
108
Structure of
Ω
108
Topological description of surgery
109
Lecture
24
Ш
Geometric surgery on
a Ricci
flow 111
Surgery (refined)
112
The standard solution
112
Lecture
25 115
Existence of
Ricci
flow with surgery defined for all time:
the statement and outline of proof
115
Noncollapsing
117
Lecture
26 121
e-canonical neighborhood threshold parameter
121
Discreteness of the surgery times
123
Part
6.
Behavior as
t
—
l· oo
125
Lecture
27 127
Recap of results of previous parts
127
Normalized volume and scalar curvature at infinity
127
Lecture
28 131
Hyperbolic limits
131
Analytic results for large time
132
Lecture
29 135
Permanence of the hyperbolic pieces
135
Hyperbolic towers
136
Lecture
30 139
viii CONTENTS
Incompressibility
of the boundary tori
139
Structure of Mttthin(w)
139
Lecture
31 141
The relative version of the Geometrization Conjecture
141
Proof that the theorem implies Geometrization Conjecture
142
Study of Mt,thin(u>)
142
Lecture
32 145
The structure of sufficiently volume collapsed 3-manifolds
145
Gromov-Hausdorff limits
145
Alexandrov spaces
145
Structure of the p~l(xn)B(xn, p(xn)) when the limit has dimension
1 147
Structure of the p^1(x7l)B(xn, p{xn)) when the limit has dimension
2 147
The global structure
148
Bibliography
149
Ibis book is based on lectures given at Stanford
University in
2009.
The purpose of the lectures
and of the book is to give an introductory over¬
view of how to use
Ricci
flow and
Ricci
flow
with surgery to establish the
Poincaré
Conjecture
and the more general Geometrization Conjecture
for 3-dimensional manifolds. Most of the mate¬
rial is geometric and analytic in nature; a crucial
ingredient is understanding singularity devel¬
opment for
З
-dimensional
Ricci
flows and for
3-dimensional
Ricci
flows with surgery. This understanding is crucial for extending
Ricci
flows with surgery so that they are defined for all positive time. Once this result is in place,
one must study the nature of the time-slices as the time goes to inanity in order to deduce
the topological consequences.
The goal of the authors is to present the major geometric and analytic results and themes of
the subject without weighing down the presentation with too many details. This book can
be read as an introduction to more complete treatments of the same material.
|
| any_adam_object | 1 |
| author | Morgan, John W. 1946- Fong, Frederick Tsz-Ho 1983- |
| author_GND | (DE-588)129352446 (DE-588)142438731 |
| author_facet | Morgan, John W. 1946- Fong, Frederick Tsz-Ho 1983- |
| author_role | aut aut |
| author_sort | Morgan, John W. 1946- |
| author_variant | j w m jw jwm f t h f fth fthf |
| building | Verbundindex |
| bvnumber | BV036126606 |
| classification_rvk | SI 165 SK 370 |
| ctrlnum | (OCoLC)699663705 (DE-599)BVBBV036126606 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV036126606 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T22:37:28Z |
| institution | BVB |
| isbn | 9780821849637 |
| language | English |
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| physical | ix, 150 Seiten Diagramme |
| publishDate | 2010 |
| publishDateSearch | 2010 |
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| publisher | American Mathematical Society |
| record_format | marc |
| series | University lecture series |
| series2 | University lecture series |
| spelling | Morgan, John W. 1946- Verfasser (DE-588)129352446 aut Ricci flow and geometrization of 3-manifolds John W. Morgan ; Frederick Tsz-Ho Fong Providence, Rhode Island American Mathematical Society [2010] © 2010 ix, 150 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier University lecture series Volume 53 Topologische Mannigfaltigkeit (DE-588)4185712-4 gnd rswk-swf Ricci-Fluss (DE-588)7531847-7 gnd rswk-swf Topologische Mannigfaltigkeit (DE-588)4185712-4 s Ricci-Fluss (DE-588)7531847-7 s DE-604 Fong, Frederick Tsz-Ho 1983- Verfasser (DE-588)142438731 aut Erscheint auch als Online-Ausgabe 978-1-4704-1648-5 University lecture series Volume 53 (DE-604)BV004153846 53 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020209005&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020209005&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Morgan, John W. 1946- Fong, Frederick Tsz-Ho 1983- Ricci flow and geometrization of 3-manifolds University lecture series Topologische Mannigfaltigkeit (DE-588)4185712-4 gnd Ricci-Fluss (DE-588)7531847-7 gnd |
| subject_GND | (DE-588)4185712-4 (DE-588)7531847-7 |
| title | Ricci flow and geometrization of 3-manifolds |
| title_auth | Ricci flow and geometrization of 3-manifolds |
| title_exact_search | Ricci flow and geometrization of 3-manifolds |
| title_full | Ricci flow and geometrization of 3-manifolds John W. Morgan ; Frederick Tsz-Ho Fong |
| title_fullStr | Ricci flow and geometrization of 3-manifolds John W. Morgan ; Frederick Tsz-Ho Fong |
| title_full_unstemmed | Ricci flow and geometrization of 3-manifolds John W. Morgan ; Frederick Tsz-Ho Fong |
| title_short | Ricci flow and geometrization of 3-manifolds |
| title_sort | ricci flow and geometrization of 3 manifolds |
| topic | Topologische Mannigfaltigkeit (DE-588)4185712-4 gnd Ricci-Fluss (DE-588)7531847-7 gnd |
| topic_facet | Topologische Mannigfaltigkeit Ricci-Fluss |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020209005&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020209005&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV004153846 |
| work_keys_str_mv | AT morganjohnw ricciflowandgeometrizationof3manifolds AT fongfredericktszho ricciflowandgeometrizationof3manifolds |