The Ricci flow in Riemannian geometry: a complete proof of the differentiable 1/4-pinching sphere theorem
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2011
|
| Series: | Lecture notes in mathematics
2011 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Literaturangaben |
| Physical Description: | XVII, 296 S. graph. Darst. 24 cm |
| ISBN: | 9783642162855 |
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| 245 | 1 | 0 | |a The Ricci flow in Riemannian geometry |b a complete proof of the differentiable 1/4-pinching sphere theorem |c Ben Andrews ; Christopher Hopper |
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Record in the Search Index
| _version_ | 1805095147459837952 |
|---|---|
| adam_text |
IMAGE 1
CONTENTS
1 INTRODUCTION 1
1.1 MANIFOLDS WITH CONSTANT SECTIONAL CURVATURE 1
1.2 THE TOPOLOGICAL SPHERE THEOREM 2
1.2.1 REMARKS ON THE CLASSICAL PROOF 3
1.2.2 MANIFOLDS WITH POSITIVE ISOTROPIE CURVATURE 4 1.2.3 A QUESTION OF
OPTIMALITY 4
1.3 THE DIFFERENTIABLE SPHERE THEOREM 5
1.3.1 THE RICCI FLOW 5
1.3.2 RICCI FLOW IN HIGHER DIMENSIONS 7
1.4 WHERE TO NEXT? 8
2 BACKGROUND MATERIAL 11
2.1 SMOOTH MANIFOLDS 11
2.1.1 TANGENT SPACE 12
2.2 VECTOR BUNDLES 12
2.2.1 SUBBUNDLES 14
2.2.2 FRAMEBUNDLES 14
2.3 TENSORS 15
2.3.1 TENSOR PRODUCTS 16
2.3.2 TENSOR CONTRACTIONS 17
2.3.3 TENSOR BUNDLES AND TENSOR FIELDS 18
2.3.4 DUALBUNDLES 19
2.3.5 TENSOR PRODUCTS OF BUNDLES 19
2.3.6 A TEST FOR TENSORALITY 19
2.4 METRIC TENSORS 20
2.4.1 RIEMANNIAN METRICS 21
2.4.2 THE PRODUCT M E T R I C. 22
2.4.3 METRIC CONTRACTIONS 22
2.4.4 METRICS ON BUNDLES 22
2.4.5 METRIC ON DUAL BUNDLES 23
2.4.6 METRIC ON TENSOR PRODUCT BUNDLES 23
2.5 CONNECTIONS 23
2.5.1 COVARIANT DERIVATIVE OF TENSOR FIELDS 25
2.5.2 THE SECOND COVARIANT DERIVATIVE OF TENSOR FIELDS. 27 XI
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1006508813
DIGITALISIERT DURCH
IMAGE 2
CONTENTS
2.5.3 CONNECTIONS ON DUAL AND TENSOR PRODUCT BUNDLES . 28 2.5.4 THE
LEVI-CIVITA CONNECTION 29
2.6 CONNECTION LAPLACIAN 30
2.7 CURVATURE 30
2.7.1 CURVATURE ON VECTOR BUNDLES 30
2.7.2 CURVATURE ON DUAL AND TENSOR PRODUCT BUNDLES 31
2.7.3 CURVATURE ON THE TENSOR BUNDLE 31
2.7.4 RIEMANNIAN CURVATURE 33
2.7.5 RICCI AND SCALAR CURVATURE 33
2.7.6 SECTIONAL CURVATURE 35
2.7.7 BERGER'S LEMMA 36
2.8 PULLBACK BUNDLE STRUCTURE 37
2.8.1 RESTRICTIONS 37
2.8.2 PUSHFORWARDS 38
2.8.3 PULLBACKS OF TENSORS 38
2.8.4 THE PULLBACK CONNECTION 39
2.8.5 PARALLEL TRANSPORT 40
2.8.6 PRODUCT MANIFOLDS' TANGENT SPACE DECOMPOSITION . 41 2.8.7
CONNECTIONS AND METRICS ON SUBBUNDLES 42
2.8.8 THE TAYLOR EXPANSION OF A RIEMANNIAN METRIC 43 2.9 INTEGRATION AND
DIVERGENCE THEOREMS 45
2.9.1 REMARKS ON THE DIVERGENCE EXPRESSION 46
HARMONIC MAPPINGS 49
3.1 GLOBAL EXISTENCE OF GEODESIES 49
3.2 HARMONIC MAP HEAT FLOW 53
3.2.1 GRADIENT FLOW OF E 54
3.2.2 EVOLUTION OF THE ENERGY DENSITY 55
3.2.3 ENERGY DENSITY BOUNDS 56
3.2.4 HIGHER REGULARITY 58
3.2.5 STABILITY LEMMA OF HARTMAN 59
3.2.6 CONVERGENCE TO A HARMONIC MAP 61
3.2.7 FURTHER RESULTS 62
EVOLUTION OF THE CURVATURE 63
4.1 INTRODUCING THE RICCI FLOW 63
4.1.1 EXACT SOLUTIONS 64
4.1.2 DIFFEOMORPHISM INVARIANCE 65
4.1.3 PARABOLIC RESCALING OF THE RICCI FLOW 66
4.2 THE LAPLACIAN OF CURVATURE 66
4.2.1 QUADRATIC CURVATURE TENSOR 67
4.2.2 CALCULATING THE CONNECTION LAPLACIAN ARIJKE 67
IMAGE 3
CONTENTS
4.3 METRIC VARIATION FORMULAS C9
4.3.1 INTERPRETING THE TIME DERIVATIVE 69
4.3.2 VARIATION FORMULAS OF THE CURVATURE 71
4.4 EVOLUTION OF THE CURVATURE UNDER THE RICCI FLOW 75
4.5 A CLOSER LOOK AT THE CURVATURE TENSOR 78
4.5.1 KULKARNI-NOMIZU PRODUCT 79
4.5.2 WEYL CURVATURE TENSOR 80
4.5.3 SPHERE THEOREM OF HUISKEN-MARGERIN-NISHIKAWA . . 81
SHORT-TIME EXISTENCE 83
5.1 THE SYMBOL 83
5.1.1 LINEAR DIFFERENTIAL OPERATORS 83
5.1.2 NONLINEAR DIFFERENTIAL OPERATORS 85
5.2 THE LINEARISATION OF THE RICCI TENSOR 86
5.3 ELLIPTICITY AND THE BIANCHI IDENTITIES 86
5.3.1 DIFFEOMORPHISM INVARIANCE OF CURVATURE AND THE BIANCHI IDENTITIES
88
5.4 DETURCK'S TRICK 90
5.4.1 MOTIVATION 90
5.4.2 RELATING RICCI-DETURCK FLOW TO RICCI FLOW 92
UHLENBECK'S TRICK 97
6.1 ABSTRACT BUNDLE APPROACH 97
6.2 ORTHONORMAL FRAME APPROACH 98
6.2.1 THE FRAME BUNDLE 99
6.2.2 TIME-DEPENDENT FRAME BUNDLES AND THE RICCI FLOW 101
6.3 TIME-DEPENDENT METRICS AND VECTOR BUNDLES OVER M XR 104
6.3.1 SPATIAL TANGENT BUNDLE AND TIME-DEPENDENT METRICS 104
6.3.2 ALTERNATIVE DERIVATION OF THE EVOLUTION OF CURVATURE EQUATION 107
THE WEAK MAXIMUM PRINCIPLE 115
7.1 ELEMENTARY ANALYSIS 115
7.2 SCALAR MAXIMUM PRINCIPLE 116
7.2.1 LOWER BOUNDS ON THE SCALAR CURVATURE 118
7.2.2 DOUBLING-TIME ESTIMATES 118
7.3 MAXIMUM PRINCIPLE FOR SYMMETRIC 2-TENSORS 119
7.4 VECTOR BUNDLE MAXIMUM PRINCIPLE 120
7.4.1 STATEMENT OF MAXIMUM PRINCIPLE 121
7.5 APPLICATIONS OF THE VECTOR BUNDLE MAXIMUM PRINCIPLE 125 7.5.1
MAXIMUM PRINCIPLE FOR SYMMETRIC 2-TENSORS REVISITED 125
IMAGE 4
XIV CONTENTS
7.5.2 REACTION-DIFFUSION EQUATION APPLICATIONS 126 7.5.3 APPLICATIONS TO
THE RICCI FLOW WHEN N = 3 129
8 REGULARITY AND LONG-TIME EXISTENCE 137
8.1 REGULARITY: THE GLOBAL SHI ESTIMATES 137
8.2 LONG-TIME EXISTENCE 140
9 THE COMPACTNESS THEOREM FOR RIEMANNIAN MANIFOLDS 145 9.1 DIFFERENT
NOTIONS OF CONVERGENCE 145
9.1.1 CONVERGENCE OF CONTINUOUS FUNCTIONS 146
9.1.2 THE SPACE OF C-FUNCTIONS AND THE C P -NORM 146 9.1.3 CONVERGENCE
OF A SEQUENCE OF SECTIONS OF A BUNDLE. 147 9.2 CHEEGER-GROMOV
CONVERGENCE 148
9.2.1 EXPANDING SPHERE EXAMPLE 149
9.2.2 THE ROSENAU METRICS 150
9.3 STATEMENT OF THE COMPACTNESS THEOREM 153
9.3.1 STATEMENT OF THE COMPACTNESS THEOREM FOR FLOWS . 155 9.4 PROOF OF
THE COMPACTNESS THEOREM FOR FLOWS 155
9.4.1 THE ARZELA-ASCOLI THEOREM 156
9.4.2 THE PROOF 157
9.5 BLOWING UP OF SINGULARITIES 158
10 THE .T 7 -FUNCTIONAL AND GRADIENT FLOWS 161
10.1 INTRODUCING THE GRADIENT FLOW FORMULATION 161
10.2 EINSTEIN-HILBERT FUNCTIONAL 163
10.3 ^-FUNCTIONAL 164
10.4 GRADIENT FLOW OF J 7 * AND ASSOCIATED COUPLED EQUATIONS- 166 10.4.1
COUPLED SYSTEMS AND THE RICCI FLOW 168
10.4.2 MONOTONICITY OF T FROM THE MONOTONICITY CAT* 170
11 THE W-FUNCTIONAL AND LOCAL NONCOLLAPSING 173
11.1 ENTROPY W-FUNCTIONAL 173
11.2 GRADIENT FLOW OF W AND MONOTONICITY 175
11.2.1 MONOTONICITY OF W FROM A POINTWISE ESTIMATE 177 11.3
/X-FUNCTIONAL 177
11.4 LOCAL NONCOLLAPSING THEOREM 181
11.4.1 LOCAL NONCOLLAPSING IMPLIES INJECTIVITY RADIUS BOUNDS 186
11.5 THE BLOW-UP OF SINGULARITIES AND LOCAL NONCOLLAPSING 187 11.6
REMARKS CONCERNING PEREL'MAN'S MOTIVATION FROM PHYSICS 189
IMAGE 5
CONTENTS XV
12 AN ALGEBRAIC IDENTITY FOR CURVATURE OPERATORS 193
12.1 A CLOSER LOOK AT TENSOR BUNDLES 194
12.1.1 INVARIANT TENSOR BUNDLES 194
12.1.2 CONSTRUCTING SUBSETS IN INVARIANT SUBBUNDLES 197 12.1.3 CHECKING
THAT THE VECTOR FIELD POINTS INTO THE SET 198
12.2 ALGEBRAIC CURVATURE OPERATORS 199
12.2.1 INTERPRETING THE REACTION TERMS 200
12.2.2 ALGEBRAIC RELATIONSHIPS AND GENERALISATIONS 204 12.3
DECOMPOSITION OF ALGEBRAIC CURVATURE OPERATORS 208 12.3.1 SCHUR'S LEMMA
209
12.3.2 THE Q-OPERATOR AND THE WEYL SUBSPACE 211
12.3.3 ALGEBRAIC LEMMAS OF BOEHM AND WILKING 212 12.4 A FAMILY OF
TRANSFORMATIONS FOR THE RICCI FLOW 215
13 THE CONE CONSTRUCTION OF BOEHM AND WILKING 223
13.1 NEW INVARIANT SETS 223
13.1.1 INITIAL CONE ASSUMPTIONS 224
13.2 GENERALISED PINCHING SETS 230
13.2.1 GENERALISED PINCHING SET EXISTENCE THEOREM 231
14 PRESERVING POSITIVE ISOTROPIE CURVATURE 235
14.1 POSITIVE ISOTROPIE CURVATURE 236
14.2 THE 1/4-PINCHING CONDITION AND PIC 238
14.2.1 THE CONE C PICK 238
14.2.2 AN ALGEBRAIC CHARACTERISATION OF THE CONE CPIC 2 * * * 241 14.3
PIC IS PRESERVED BY THE RICCI FLOW 243
14.3.1 INEQUALITIES FROM THE SECOND DERIVATIVE TEST 245 14.4 PCSC IS
PRESERVED BY THE RICCI FLOW 252
14.4.1 THE MOK LEMMA 252
14.4.2 PRESERVATION OF PCSC PROOF 253
14.4.3 RELATING PCSC TO PIC 254
14.5 PRESERVING PIC USING THE COMPLEXIFICATION 255
15 THE FINAL ARGUMENT 259
15.1 PROOF OF THE SPHERE THEOREM 259
15.2 REFINED CONVERGENCE RESULT * 260
15.2.1 A PRESERVED SET BETWEEN C PICL AND C P IC 2 262
15.2.2 A PINCHING SET ARGUMENT 265
IMAGE 6
XVI CONTENTS
A GATEAUX AND FRECHET DIFFERENTIABILITY 271
A.I PROPERTIES OF THE GATEAUX DERIVATIVE 272
B CONES, CONVEX SETS AND SUPPORT FUNCTIONS 275
B.I CONVEX SETS 275
B.2 SUPPORT FUNCTIONS 275
B.3 THE DISTANCE FROM A CONVEX SET 276
B.4 TANGENT AND NORMAL CONES 277
B.5 CONVEX SETS DEFINED BY INEQUALITIES 277
C CANONICALLY IDENTIFYING TENSOR SPACES WITH LIE ALGEBRAS . 281 C.I LIE
ALGEBRAS 281
C.2 TENSOR SPACES AS LIE ALGEBRAS 282
C.3 THE SPACE OF SECOND EXTERIOR POWERS AS A LIE ALGEBRA 282 C.3.1 THE
SPACE FT 2 V* AS A LIE ALGEBRA 284
REFERENCES 285
INDEX 293 |
| any_adam_object | 1 |
| author | Andrews, Ben Hopper, Christopher |
| author_GND | (DE-588)143084186 (DE-588)143084194 |
| author_facet | Andrews, Ben Hopper, Christopher |
| author_role | aut aut |
| author_sort | Andrews, Ben |
| author_variant | b a ba c h ch |
| building | Verbundindex |
| bvnumber | BV036887961 |
| classification_rvk | SI 850 |
| classification_tum | MAT 537f |
| ctrlnum | (OCoLC)694879407 (DE-599)DNB1006508813 |
| dewey-full | 516.362 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.362 |
| dewey-search | 516.362 |
| dewey-sort | 3516.362 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV036887961 |
| illustrated | Illustrated |
| indexdate | 2024-07-20T10:54:39Z |
| institution | BVB |
| isbn | 9783642162855 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020803203 |
| oclc_num | 694879407 |
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| owner_facet | DE-824 DE-91G DE-BY-TUM DE-11 DE-83 DE-29T DE-188 DE-19 DE-BY-UBM |
| physical | XVII, 296 S. graph. Darst. 24 cm |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Andrews, Ben Verfasser (DE-588)143084186 aut The Ricci flow in Riemannian geometry a complete proof of the differentiable 1/4-pinching sphere theorem Ben Andrews ; Christopher Hopper Berlin [u.a.] Springer 2011 XVII, 296 S. graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 2011 Literaturangaben Ricci-Fluss (DE-588)7531847-7 gnd rswk-swf Riemannsche Geometrie (DE-588)4128462-8 gnd rswk-swf Ricci-Fluss (DE-588)7531847-7 s Riemannsche Geometrie (DE-588)4128462-8 s DE-604 Hopper, Christopher Verfasser (DE-588)143084194 aut Erscheint auch als Online-Ausgabe 978-3-642-16286-2 Lecture notes in mathematics 2011 (DE-604)BV000676446 2011 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3532122&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020803203&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Andrews, Ben Hopper, Christopher The Ricci flow in Riemannian geometry a complete proof of the differentiable 1/4-pinching sphere theorem Lecture notes in mathematics Ricci-Fluss (DE-588)7531847-7 gnd Riemannsche Geometrie (DE-588)4128462-8 gnd |
| subject_GND | (DE-588)7531847-7 (DE-588)4128462-8 |
| title | The Ricci flow in Riemannian geometry a complete proof of the differentiable 1/4-pinching sphere theorem |
| title_auth | The Ricci flow in Riemannian geometry a complete proof of the differentiable 1/4-pinching sphere theorem |
| title_exact_search | The Ricci flow in Riemannian geometry a complete proof of the differentiable 1/4-pinching sphere theorem |
| title_full | The Ricci flow in Riemannian geometry a complete proof of the differentiable 1/4-pinching sphere theorem Ben Andrews ; Christopher Hopper |
| title_fullStr | The Ricci flow in Riemannian geometry a complete proof of the differentiable 1/4-pinching sphere theorem Ben Andrews ; Christopher Hopper |
| title_full_unstemmed | The Ricci flow in Riemannian geometry a complete proof of the differentiable 1/4-pinching sphere theorem Ben Andrews ; Christopher Hopper |
| title_short | The Ricci flow in Riemannian geometry |
| title_sort | the ricci flow in riemannian geometry a complete proof of the differentiable 1 4 pinching sphere theorem |
| title_sub | a complete proof of the differentiable 1/4-pinching sphere theorem |
| topic | Ricci-Fluss (DE-588)7531847-7 gnd Riemannsche Geometrie (DE-588)4128462-8 gnd |
| topic_facet | Ricci-Fluss Riemannsche Geometrie |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3532122&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020803203&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT andrewsben thericciflowinriemanniangeometryacompleteproofofthedifferentiable14pinchingspheretheorem AT hopperchristopher thericciflowinriemanniangeometryacompleteproofofthedifferentiable14pinchingspheretheorem |