Ricci flow and the Poincaré conjecture:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Math. Soc.
2007
|
| Series: | Clay mathematics monographs
3 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 515 - 518 |
| Physical Description: | XLII, , 521 S. Ill., graph. Darst. 27 cm |
| ISBN: | 9780821843284 0821843281 |
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 the Poincaré conjecture |c John Morgan ; Gang Tian |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2007 | |
| 300 | |a XLII, , 521 S. |b Ill., graph. Darst. |c 27 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Clay mathematics monographs |v 3 | |
| 500 | |a Literaturverz. S. 515 - 518 | ||
| 650 | 7 | |a Differentiaalmeetkunde |2 gtt | |
| 650 | 7 | |a Poincaré groups |2 gtt | |
| 650 | 7 | |a Pseudo Riemannian-manifolds |2 gtt | |
| 650 | 7 | |a Riemann-vlakken |2 gtt | |
| 650 | 4 | |a Ricci flow | |
| 650 | 4 | |a Poincaré conjecture | |
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Record in the Search Index
| _version_ | 1804137744272719872 |
|---|---|
| adam_text | Contents
Introduction
ix
1.
Overview of
Perelman
s argument x
2.
Background material from Riemannian geometry
xvi
3.
Background material from
Ricci
flow
xix
4.
Perelman s advances
xxv
5.
The standard solution and the surgery process
xxxi
6.
Extending
Ricci
flows with surgery
xxxiv
7.
Finite-time extinction
xxxvii
8.
Acknowledgements
xl
9.
List of related papers
xlii
Part
1.
Background from Riemannian Geometry and
Ricci
flow
1
Chapter
1.
Preliminaries from Riemannian geometry
3
1.
Riemannian metrics and the Levi-Civita connection
3
2.
Curvature of a Riemannian manifold
5
3.
Geodesies and the exponential map
10
4.
Computations in Gaussian normal coordinates
16
5.
Basic curvature comparison results
18
6.
Local volume and the injectivity radius
19
Chapter
2.
Manifolds of non-negative curvature
21
1.
Busemann functions
21
2.
Comparison results in non-negative curvature
23
3.
The soul theorem
24
4.
Ends of a manifold
27
5.
The splitting theorem
28
6.
e-necks
30
7.
Forward difference quotients
33
Chapter
3.
Basics of
Ricci
flow
35
1.
The definition of
Ricci
flow
35
2.
Some exact solutions to the
Ricci
flow
36
3.
Local existence and uniqueness
39
4.
Evolution of curvatures
41
vi
CONTENTS
5.
Curvature
evolution in an evolving orthonormal frame
42
6.
Variation of distance under
Ricci
flow
45
7.
Shi s derivative estimates
50
8.
Generalized
Ricci
flows
59
Chapter
4.
The maximum principle
63
1.
Maximum principle for scalar curvature
63
2.
The maximum principle for tensors
65
3.
Applications of the maximum principle
67
4.
The strong maximum principle for curvature
69
5.
Pinching toward positive curvature
75
Chapter
5.
Convergence results for
Ricci
flow
83
1.
Geometric convergence of Riemannian manifolds
83
2.
Geometric convergence of
Ricci
flows
90
3.
Gromov-Hausdorff convergence
92
4.
Blow-up limits
99
5.
Splitting limits at infinity
100
Part
2.
Perelman s length function and its applications
103
Chapter
6.
A comparison geometry approach to the
Ricci
flow
105
1.
¿-length and ¿-geodesies
105
2.
The ¿-exponential map and its first-order properties
112
3.
Minimizing ¿-geodesies and the injectivity domain
116
4.
Second-order differential inequalities for LT and
L
Tx
119
5.
Reduced length
129
6.
Local Lipschitz estimates for lx
133
7.
Reduced volume
140
Chapter
7.
Complete
Ricci
flows of bounded curvature
149
1.
The functions Lx and lx
149
2.
A bound for
min
Ц,
152
3.
Reduced volume
164
Chapter
8.
Non-collapsed results
169
1.
A non-collapsing result for generalized
Ricci
flows
169
2.
Application to compact
Ricci
flows
176
Chapter
9.
re-non-collapsed ancient solutions
179
1.
Preliminaries
179
2.
The asymptotic gradient shrinking soliton for «-solutions
183
3.
Splitting results at infinity
203
4.
Classification of gradient shrinking
solitone
206
5.
Universal
к
220
6.
Asymptotic volume
221
CONTENTS
vii
7. Compactness
of the space of
3-dimensional
к
-solutions
225
8.
Qualitative description of K-solutions
230
Chapter
10.
Bounded curvature at bounded distance
245
1.
Pinching toward positive: the definitions
245
2.
The statement of the theorem
245
3.
The incomplete geometric limit
247
4.
Cone limits near the end
E
for rescalings of Uoo
255
5.
Comparison of the two types of limits
263
6.
The final contradiction
265
Chapter
11.
Geometric limits of generalized
Ricci
flows
267
1.
A smooth blow-up limit defined for a small time
267
2.
Long-time blow-up limits
271
3.
Incomplete smooth limits at singular times
279
4.
Existence of strong ¿-necks sufficiently deep in a 2e-horn
287
Chapter
12.
The standard solution
293
1.
The initial metric
293
2.
Standard
Ricci
flows: The statement
295
3.
Existence of a standard flow
296
4.
Completeness, positive curvature, and asymptotic behavior
297
5.
Standard solutions are rotationally symmetric
300
6.
Uniqueness
306
7.
Solution of the harmonic map flow
308
8.
Completion of the proof of uniqueness
322
9.
Some corollaries
325
Part
3.
Ricci
flow with surgery
329
Chapter
13.
Surgery on a ¿-neck
331
1.
Notation and the statement of the result
331
2.
Preliminary computations
334
3.
The proof of Theorem
13.2 339
4.
Other properties of the result of surgery
341
Chapter
14.
Ricci
Flow with surgery: the definition
343
1.
Surgery space-time
343
2.
The generalized
Ricci
flow equation
348
Chapter
15.
Controlled
Ricci
flows with surgery
353
1.
Gluing together evolving necks
353
2.
Topologica!
consequences of Assumptions
(1) - (7) 356
3.
Further conditions on surgery
359
4.
The process of surgery
361
5.
Statements about the existence of
Ricci
flow with surgery
362
viii
CONTENTS
6.
Outline of the proof of Theorem
15.9 365
Chapter
16.
Proof of non-collapsing
367
1.
The statement of the non-collapsing result
367
2.
The proof of non-collapsing when R(x)
=
r~2 with
r
<
Гј
+i
368
3.
Minimizing
Zľ-geodesics
exist when R(x)
<
rŢ^:
the statement
368
4.
Evolution of neighborhoods of surgery caps
369
5.
A length estimate
375
6.
Completion of the proof of Proposition
16.1 391
Chapter
17.
Completion of the proof of Theorem
15.9 395
1.
Proof of the strong canonical neighborhood assumption
395
2.
Surgery times don t accumulate
408
Part
4.
Completion of the proof of the
Poincaré
Conjecture
413
Chapter
18.
Finite-time extinction
415
1.
The result
415
2.
Disappearance of components with non-trivial
тг2
420
3.
Components with non-trivial
7Г3
429
4.
First steps in the proof of Proposition
18.18 432
Chapter
19.
Completion of the Proof of Proposition
18.24 437
1.
Curve-shrinking
437
2.
Basic estimates for curve-shrinking
441
3.
Ramp solutions in
M x S1
445
4.
Approximating the original family
Γ
449
5.
The case of a single
с
Є
S2
453
6.
The completion of the proof of Proposition
18.24 461
7.
Proof of Lemma
19.31:
annuii
of small area
464
8.
Proof of the first inequality in Lemma
19.24 481
Appendix. 3-manifolds covered by canonical neighborhoods
497
1.
Shortening curves
497
2.
The geometry of an e-neck
497
3.
Overlapping e-necks
502
4.
Regions covered by e-necks and (C, e)-caps
504
5.
Subsets of the union of cores of (C,
е)-сарѕ
and e-necks.
508
Bibliography
515
Index
519
|
| adam_txt |
Contents
Introduction
ix
1.
Overview of
Perelman
's argument x
2.
Background material from Riemannian geometry
xvi
3.
Background material from
Ricci
flow
xix
4.
Perelman's advances
xxv
5.
The standard solution and the surgery process
xxxi
6.
Extending
Ricci
flows with surgery
xxxiv
7.
Finite-time extinction
xxxvii
8.
Acknowledgements
xl
9.
List of related papers
xlii
Part
1.
Background from Riemannian Geometry and
Ricci
flow
1
Chapter
1.
Preliminaries from Riemannian geometry
3
1.
Riemannian metrics and the Levi-Civita connection
3
2.
Curvature of a Riemannian manifold
5
3.
Geodesies and the exponential map
10
4.
Computations in Gaussian normal coordinates
16
5.
Basic curvature comparison results
18
6.
Local volume and the injectivity radius
19
Chapter
2.
Manifolds of non-negative curvature
21
1.
Busemann functions
21
2.
Comparison results in non-negative curvature
23
3.
The soul theorem
24
4.
Ends of a manifold
27
5.
The splitting theorem
28
6.
e-necks
30
7.
Forward difference quotients
33
Chapter
3.
Basics of
Ricci
flow
35
1.
The definition of
Ricci
flow
35
2.
Some exact solutions to the
Ricci
flow
36
3.
Local existence and uniqueness
39
4.
Evolution of curvatures
41
vi
CONTENTS
5.
Curvature
evolution in an evolving orthonormal frame
42
6.
Variation of distance under
Ricci
flow
45
7.
Shi's derivative estimates
50
8.
Generalized
Ricci
flows
59
Chapter
4.
The maximum principle
63
1.
Maximum principle for scalar curvature
63
2.
The maximum principle for tensors
65
3.
Applications of the maximum principle
67
4.
The strong maximum principle for curvature
69
5.
Pinching toward positive curvature
75
Chapter
5.
Convergence results for
Ricci
flow
83
1.
Geometric convergence of Riemannian manifolds
83
2.
Geometric convergence of
Ricci
flows
90
3.
Gromov-Hausdorff convergence
92
4.
Blow-up limits
99
5.
Splitting limits at infinity
100
Part
2.
Perelman's length function and its applications
103
Chapter
6.
A comparison geometry approach to the
Ricci
flow
105
1.
¿-length and ¿-geodesies
105
2.
The ¿-exponential map and its first-order properties
112
3.
Minimizing ¿-geodesies and the injectivity domain
116
4.
Second-order differential inequalities for LT and
L
Tx
119
5.
Reduced length
129
6.
Local Lipschitz estimates for lx
133
7.
Reduced volume
140
Chapter
7.
Complete
Ricci
flows of bounded curvature
149
1.
The functions Lx and lx
149
2.
A bound for
min
Ц,
152
3.
Reduced volume
164
Chapter
8.
Non-collapsed results
169
1.
A non-collapsing result for generalized
Ricci
flows
169
2.
Application to compact
Ricci
flows
176
Chapter
9.
re-non-collapsed ancient solutions
179
1.
Preliminaries
179
2.
The asymptotic gradient shrinking soliton for «-solutions
183
3.
Splitting results at infinity
203
4.
Classification of gradient shrinking
solitone
206
5.
Universal
к
220
6.
Asymptotic volume
221
CONTENTS
vii
7. Compactness
of the space of
3-dimensional
к
-solutions
225
8.
Qualitative description of K-solutions
230
Chapter
10.
Bounded curvature at bounded distance
245
1.
Pinching toward positive: the definitions
245
2.
The statement of the theorem
245
3.
The incomplete geometric limit
247
4.
Cone limits near the end
E
for rescalings of Uoo
255
5.
Comparison of the two types of limits
263
6.
The final contradiction
265
Chapter
11.
Geometric limits of generalized
Ricci
flows
267
1.
A smooth blow-up limit defined for a small time
267
2.
Long-time blow-up limits
271
3.
Incomplete smooth limits at singular times
279
4.
Existence of strong ¿-necks sufficiently deep in a 2e-horn
287
Chapter
12.
The standard solution
293
1.
The initial metric
293
2.
Standard
Ricci
flows: The statement
295
3.
Existence of a standard flow
296
4.
Completeness, positive curvature, and asymptotic behavior
297
5.
Standard solutions are rotationally symmetric
300
6.
Uniqueness
306
7.
Solution of the harmonic map flow
308
8.
Completion of the proof of uniqueness
322
9.
Some corollaries
325
Part
3.
Ricci
flow with surgery
329
Chapter
13.
Surgery on a ¿-neck
331
1.
Notation and the statement of the result
331
2.
Preliminary computations
334
3.
The proof of Theorem
13.2 339
4.
Other properties of the result of surgery
341
Chapter
14.
Ricci
Flow with surgery: the definition
343
1.
Surgery space-time
343
2.
The generalized
Ricci
flow equation
348
Chapter
15.
Controlled
Ricci
flows with surgery
353
1.
Gluing together evolving necks
353
2.
Topologica!
consequences of Assumptions
(1) - (7) 356
3.
Further conditions on surgery
359
4.
The process of surgery
361
5.
Statements about the existence of
Ricci
flow with surgery
362
viii
CONTENTS
6.
Outline of the proof of Theorem
15.9 365
Chapter
16.
Proof of non-collapsing
367
1.
The statement of the non-collapsing result
367
2.
The proof of non-collapsing when R(x)
=
r~2 with
r
<
Гј
+i
368
3.
Minimizing
Zľ-geodesics
exist when R(x)
<
rŢ^:
the statement
368
4.
Evolution of neighborhoods of surgery caps
369
5.
A length estimate
375
6.
Completion of the proof of Proposition
16.1 391
Chapter
17.
Completion of the proof of Theorem
15.9 395
1.
Proof of the strong canonical neighborhood assumption
395
2.
Surgery times don't accumulate
408
Part
4.
Completion of the proof of the
Poincaré
Conjecture
413
Chapter
18.
Finite-time extinction
415
1.
The result
415
2.
Disappearance of components with non-trivial
тг2
420
3.
Components with non-trivial
7Г3
429
4.
First steps in the proof of Proposition
18.18 432
Chapter
19.
Completion of the Proof of Proposition
18.24 437
1.
Curve-shrinking
437
2.
Basic estimates for curve-shrinking
441
3.
Ramp solutions in
M x S1
445
4.
Approximating the original family
Γ
449
5.
The case of a single
с
Є
S2
453
6.
The completion of the proof of Proposition
18.24 461
7.
Proof of Lemma
19.31:
annuii
of small area
464
8.
Proof of the first inequality in Lemma
19.24 481
Appendix. 3-manifolds covered by canonical neighborhoods
497
1.
Shortening curves
497
2.
The geometry of an e-neck
497
3.
Overlapping e-necks
502
4.
Regions covered by e-necks and (C, e)-caps
504
5.
Subsets of the union of cores of (C,
е)-сарѕ
and e-necks.
508
Bibliography
515
Index
519 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Morgan, John W. 1946- Gang, Tian 1958- |
| author_GND | (DE-588)129352446 (DE-588)11804768X |
| author_facet | Morgan, John W. 1946- Gang, Tian 1958- |
| author_role | aut aut |
| author_sort | Morgan, John W. 1946- |
| author_variant | j w m jw jwm t g tg |
| building | Verbundindex |
| bvnumber | BV023374823 |
| callnumber-first | Q - Science |
| callnumber-label | QA670 |
| callnumber-raw | QA670 |
| callnumber-search | QA670 |
| callnumber-sort | QA 3670 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 320 SK 370 |
| ctrlnum | (OCoLC)145147251 (DE-599)BVBBV023374823 |
| dewey-full | 516.3/62 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/62 |
| dewey-search | 516.3/62 |
| dewey-sort | 3516.3 262 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV023374823 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:14:00Z |
| indexdate | 2024-07-09T21:17:09Z |
| institution | BVB |
| isbn | 9780821843284 0821843281 |
| language | English |
| lccn | 2007062016 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016558020 |
| oclc_num | 145147251 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-703 DE-11 DE-188 DE-384 DE-83 DE-29T |
| owner_facet | DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-703 DE-11 DE-188 DE-384 DE-83 DE-29T |
| physical | XLII, , 521 S. Ill., graph. Darst. 27 cm |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | American Math. Soc. |
| record_format | marc |
| series | Clay mathematics monographs |
| series2 | Clay mathematics monographs |
| spelling | Morgan, John W. 1946- Verfasser (DE-588)129352446 aut Ricci flow and the Poincaré conjecture John Morgan ; Gang Tian Providence, RI American Math. Soc. 2007 XLII, , 521 S. Ill., graph. Darst. 27 cm txt rdacontent n rdamedia nc rdacarrier Clay mathematics monographs 3 Literaturverz. S. 515 - 518 Differentiaalmeetkunde gtt Poincaré groups gtt Pseudo Riemannian-manifolds gtt Riemann-vlakken gtt Ricci flow Poincaré conjecture Poincaré-Vermutung (DE-588)4517256-0 gnd rswk-swf Ricci-Fluss (DE-588)7531847-7 gnd rswk-swf Ricci-Fluss (DE-588)7531847-7 s Poincaré-Vermutung (DE-588)4517256-0 s DE-604 Gang, Tian 1958- Verfasser (DE-588)11804768X aut Clay mathematics monographs 3 (DE-604)BV017693220 3 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016558020&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Morgan, John W. 1946- Gang, Tian 1958- Ricci flow and the Poincaré conjecture Clay mathematics monographs Differentiaalmeetkunde gtt Poincaré groups gtt Pseudo Riemannian-manifolds gtt Riemann-vlakken gtt Ricci flow Poincaré conjecture Poincaré-Vermutung (DE-588)4517256-0 gnd Ricci-Fluss (DE-588)7531847-7 gnd |
| subject_GND | (DE-588)4517256-0 (DE-588)7531847-7 |
| title | Ricci flow and the Poincaré conjecture |
| title_auth | Ricci flow and the Poincaré conjecture |
| title_exact_search | Ricci flow and the Poincaré conjecture |
| title_exact_search_txtP | Ricci flow and the Poincaré conjecture |
| title_full | Ricci flow and the Poincaré conjecture John Morgan ; Gang Tian |
| title_fullStr | Ricci flow and the Poincaré conjecture John Morgan ; Gang Tian |
| title_full_unstemmed | Ricci flow and the Poincaré conjecture John Morgan ; Gang Tian |
| title_short | Ricci flow and the Poincaré conjecture |
| title_sort | ricci flow and the poincare conjecture |
| topic | Differentiaalmeetkunde gtt Poincaré groups gtt Pseudo Riemannian-manifolds gtt Riemann-vlakken gtt Ricci flow Poincaré conjecture Poincaré-Vermutung (DE-588)4517256-0 gnd Ricci-Fluss (DE-588)7531847-7 gnd |
| topic_facet | Differentiaalmeetkunde Poincaré groups Pseudo Riemannian-manifolds Riemann-vlakken Ricci flow Poincaré conjecture Poincaré-Vermutung Ricci-Fluss |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016558020&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV017693220 |
| work_keys_str_mv | AT morganjohnw ricciflowandthepoincareconjecture AT gangtian ricciflowandthepoincareconjecture |