| Description |
1 online resource (x, 113 pages) : illustrations |
| Series |
London Mathematical Society Lecture Note Series ; no. 325 |
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London Mathematical Society lecture note series ; no. 325.
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| Contents |
Introduction ; 1.1 Ricci flow: what is it, and from where did it come? -- 1.2 Examples and special solutions ; 1.2.1 Einstein manifolds -- 1.2.2 Ricci solitons -- 1.2.3 Parabolic rescaling of Ricci flows -- 1.3 Getting a feel for Ricci flow; 1.3.1 Two dimensions -- 1.3.2 Three dimensions -- 1.4 The topology and geometry of manifolds in low dimensions -- 1.5 Using Ricci flow to prove topological and geometric results -- |
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Riemannian geometry background ; 2.1 Notation and conventions -- 2.2 Einstein metrics --; 2.3 Deformation of geometric quantities as the Riemannian metric is deformed ; 2.3.1 The formulae -- 2.3.2 The calculations -- 2.4 Laplacian of the curvature tensor -- 2.5 Evolution of curvature and geometric quantities under Ricci flow -- 3 The maximum principle ; 3.1 Statement of the maximum principle -- 3.2 Basic control on the evolution of curvature -- 3.3 Global curvature derivative estimates -- 4 Comments on existence theory for parabolic PDE ; 4.1 Linear scalar PDE -- 4.2 The principal symbol -- 4.3 Generalisation to vector bundles -- 4.4 Properties of parabolic equations -- |
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5 Existence theory for the Ricci flow ; 5.1 Ricci flow is not parabolic -- 5.2 Short-time existence and uniqueness : the DeTurck trick -- 5.3 Curvature blow-up at finite-time singularities -- 6 Ricci flow as a gradient flow ; 6.1 Gradient of total scalar curvature and related functionals -- 6.2 The [script capital] F-functional -- 6.3 The heat operator and its conjugate -- 6.4 A gradient flow formulation -- 6.5 The classical entropy -- 6.6 The zeroth eigenvalue of -4[capital Greek]Delta + [italic capital]R -- |
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7 Compactness of Riemannian manifolds and flows ; 7.1 Convergence and compactness of manifolds -- 7.2 Convergence and compactness of flows -- 7.3 Blowing up at singularities I -- 8 Perelman's [script capital]W entropy functional ; 8.1 Definition, motivation and basic properties -- 8.2 Monotonicity of [script capital]W -- 8.3 No local volume collapse where curvature is controlled -- 8.4 Volume ratio bounds imply injectivity radius bounds -- 8.5 Blowing up at singularities II -- |
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9 Curvature pinching and preserved curvature properties under Ricci flow ; 9.1 Overview -- 9.2 The Einstein Tensor, [italic capital]E -- 9.3 Evolution of [italic capital]E under the Ricci flow -- 9.4 The Uhlenbeck trick -- 9.5 Formulae for parallel functions on vector bundles -- 9.6 An ODE-PDE theorem -- 9.7 Applications of the ODE-PDE theorem -- Appendix A. Connected sum |
| Summary |
These notes represent an updated version of a course on Hamilton’s Ricci flow that I gave at the University of Warwick in the spring of 2004. I have aimed to give an introduction to the main ideas of the subject, a large proportion of which are due to Hamilton over the period since he introduced the Ricci flow in 1982. The main difference between these notes and others which are available at the time of writing is that I follow the quite different route which is natural in the light of work of Perelman from 2002. It is now understood how to ‘blow up’ general Ricci flows near their singularities, as one is used to doing in other contexts within geometric analysis. This technique is now central to the subject and is emphasized throughout. The original lectures were delivered to a mixture of graduate students, postdocs, staff, and even some undergraduates. Generally I assumed that the audience had just completed a first course in differential geometry, and an elementary course in PDE, and were just about to embark on a more advanced course in PDE. I tried to make the lectures accessible to the general mathematician motivated by the applications of the theory to the Poincaré conjecture, and Thurston’s geometrisation conjecture |
| Notes |
Title from publishers bibliographic system (viewed 22 Dec 2011) |
| Bibliography |
Includes bibliographical references (pages 109-111), and index |
| Notes |
English |
| Subject |
Ricci flow.
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Geometry, Riemannian.
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Mathematics.
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MATHEMATICS -- Geometry -- Differential.
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Ricci flow
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| Genre/Form |
Instructional and educational works.
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Matériel d'éducation et de formation.
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| Form |
Electronic book
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| ISBN |
9780511721465 |
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0511721463 |
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9780521689472 |
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0521689473 |
|
9781107367807 |
|
1107367808 |
|
9781107362895 |
|
110736289X |
|
1139882627 |
|
9781139882620 |
|
1107372348 |
|
9781107372344 |
|
1107368510 |
|
9781107368514 |
|
