Lectures on the Ricci flow:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2006
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | London Mathematical Society lecture note series
325 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 109 - 111 |
| Beschreibung: | X, 113 S. graph. Darst. |
| ISBN: | 0521689473 9780521689472 |
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| 245 | 1 | 0 | |a Lectures on the Ricci flow |c Peter Topping |
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| 490 | 1 | |a London Mathematical Society lecture note series |v 325 | |
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents
Preface page ix
1 Introduction 1
1.1 Ricci flow: what is it, and from where
did it come? 1
1.2 Examples and special solutions 2
1.2.1 Einstein manifolds 2
1.2.2 Ricci solitons 3
1.2.3 Parabolic rescaling of Ricci flows 6
1.3 Getting a feel for Ricci flow 6
1.3.1 Two dimensions 6
1.3.2 Three dimensions 7
1.4 The topology and geometry of manifolds in
low dimensions 11
1.5 Using Ricci flow to prove topological and
geometric results 14
2 Riemannian geometry background 16
2.1 Notation and conventions 16
2.2 Einstein metrics 19
2.3 Deformation of geometric quantities as the Riemannian
metric is deformed 20
2.3.1 The formulae 20
2.3.2 The calculations 23
2.4 Laplacian of the curvature tensor 31
2.5 Evolution of curvature and geometric quantities
under Ricci flow 32
v
vi Contents
3 The maximum principle 35
3.1 Statement of the maximum principle 35
3.2 Basic control on the evolution of curvature 36
3.3 Global curvature derivative estimates 39
4 Comments on existence theory for parabolic PDE 43
4.1 Linear scalar PDE 43
4.2 The principal symbol 44
4.3 Generalisation to Vector Bundles 45
4.4 Properties of parabolic equations 47
5 Existence theory for the Ricci flow 48
5.1 Ricci flow is not parabolic 48
5.2 Short time existence and uniqueness: The DeTurck trick 49
5.3 Curvature blow up at finite time singularities 52
6 Ricci flow as a gradient flow 55
6.1 Gradient of total scalar curvature and related functionals 55
6.2 The jF functional 56
6.3 The heat operator and its conjugate 57
6.4 A gradient flow formulation 58
6.5 The classical entropy 61
6.6 The zeroth eigenvalue of —4A + R 63
7 Compactness of Riemannian manifolds and flows 65
7.1 Convergence and compactness of manifolds 65
7.2 Convergence and compactness of flows 68
7.3 Blowing up at singularities I 69
8 Perelman s W entropy functional 71
8.1 Definition, motivation and basic properties 71
8.2 Monotonicity of W 76
8.3 No local volume collapse where curvature is controlled 79
8.4 Volume ratio bounds imply injectivity radius bounds 84
8.5 Blowing up at singularities II 86
9 Curvature pinching and preserved curvature properties
under Ricci flow 88
9.1 Overview 88
9.2 The Einstein Tensor, E 88
Contents vii
9.3 Evolution of E under the Ricci flow 89
9.4 The Uhlenbeck Trick 90
9.5 Formulae for parallel functions on vector bundles 93
9.6 An ODE PDE theorem 95
9.7 Applications of the ODE PDE theorem 98
10 Three manifolds with positive Ricci curvature, and beyond 105
10.1 Hamilton s theorem 105
10.2 Beyond the case of positive Ricci curvature 107
A Connected sum 108
References 109
Index 112
|
| adam_txt |
Contents
Preface page ix
1 Introduction 1
1.1 Ricci flow: what is it, and from where
did it come? 1
1.2 Examples and special solutions 2
1.2.1 Einstein manifolds 2
1.2.2 Ricci solitons 3
1.2.3 Parabolic rescaling of Ricci flows 6
1.3 Getting a feel for Ricci flow 6
1.3.1 Two dimensions 6
1.3.2 Three dimensions 7
1.4 The topology and geometry of manifolds in
low dimensions 11
1.5 Using Ricci flow to prove topological and
geometric results 14
2 Riemannian geometry background 16
2.1 Notation and conventions 16
2.2 Einstein metrics 19
2.3 Deformation of geometric quantities as the Riemannian
metric is deformed 20
2.3.1 The formulae 20
2.3.2 The calculations 23
2.4 Laplacian of the curvature tensor 31
2.5 Evolution of curvature and geometric quantities
under Ricci flow 32
v
vi Contents
3 The maximum principle 35
3.1 Statement of the maximum principle 35
3.2 Basic control on the evolution of curvature 36
3.3 Global curvature derivative estimates 39
4 Comments on existence theory for parabolic PDE 43
4.1 Linear scalar PDE 43
4.2 The principal symbol 44
4.3 Generalisation to Vector Bundles 45
4.4 Properties of parabolic equations 47
5 Existence theory for the Ricci flow 48
5.1 Ricci flow is not parabolic 48
5.2 Short time existence and uniqueness: The DeTurck trick 49
5.3 Curvature blow up at finite time singularities 52
6 Ricci flow as a gradient flow 55
6.1 Gradient of total scalar curvature and related functionals 55
6.2 The jF functional 56
6.3 The heat operator and its conjugate 57
6.4 A gradient flow formulation 58
6.5 The classical entropy 61
6.6 The zeroth eigenvalue of —4A + R 63
7 Compactness of Riemannian manifolds and flows 65
7.1 Convergence and compactness of manifolds 65
7.2 Convergence and compactness of flows 68
7.3 Blowing up at singularities I 69
8 Perelman's W entropy functional 71
8.1 Definition, motivation and basic properties 71
8.2 Monotonicity of W 76
8.3 No local volume collapse where curvature is controlled 79
8.4 Volume ratio bounds imply injectivity radius bounds 84
8.5 Blowing up at singularities II 86
9 Curvature pinching and preserved curvature properties
under Ricci flow 88
9.1 Overview 88
9.2 The Einstein Tensor, E 88
Contents vii
9.3 Evolution of E under the Ricci flow 89
9.4 The Uhlenbeck Trick 90
9.5 Formulae for parallel functions on vector bundles 93
9.6 An ODE PDE theorem 95
9.7 Applications of the ODE PDE theorem 98
10 Three manifolds with positive Ricci curvature, and beyond 105
10.1 Hamilton's theorem 105
10.2 Beyond the case of positive Ricci curvature 107
A Connected sum 108
References 109
Index 112 |
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| spelling | Topping, Peter Verfasser aut Lectures on the Ricci flow Peter Topping 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2006 X, 113 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society lecture note series 325 Literaturverz. S. 109 - 111 Ricci-Fluss Ricci flow Mannigfaltigkeit (DE-588)4037379-4 gnd rswk-swf Dimension 3 (DE-588)4321722-9 gnd rswk-swf Ricci-Fluss (DE-588)7531847-7 gnd rswk-swf Ricci-Fluss (DE-588)7531847-7 s DE-604 Mannigfaltigkeit (DE-588)4037379-4 s Dimension 3 (DE-588)4321722-9 s London Mathematical Society lecture note series 325 (DE-604)BV000000130 325 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015010393&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Topping, Peter Lectures on the Ricci flow London Mathematical Society lecture note series Ricci-Fluss Ricci flow Mannigfaltigkeit (DE-588)4037379-4 gnd Dimension 3 (DE-588)4321722-9 gnd Ricci-Fluss (DE-588)7531847-7 gnd |
| subject_GND | (DE-588)4037379-4 (DE-588)4321722-9 (DE-588)7531847-7 |
| title | Lectures on the Ricci flow |
| title_auth | Lectures on the Ricci flow |
| title_exact_search | Lectures on the Ricci flow |
| title_exact_search_txtP | Lectures on the Ricci flow |
| title_full | Lectures on the Ricci flow Peter Topping |
| title_fullStr | Lectures on the Ricci flow Peter Topping |
| title_full_unstemmed | Lectures on the Ricci flow Peter Topping |
| title_short | Lectures on the Ricci flow |
| title_sort | lectures on the ricci flow |
| topic | Ricci-Fluss Ricci flow Mannigfaltigkeit (DE-588)4037379-4 gnd Dimension 3 (DE-588)4321722-9 gnd Ricci-Fluss (DE-588)7531847-7 gnd |
| topic_facet | Ricci-Fluss Ricci flow Mannigfaltigkeit Dimension 3 |
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