Lectures on the Ricci flow /

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...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Topping, Peter, 1971-
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Cambridge : Cambridge University Press, 2006.
Colección:London Mathematical Society lecture note series ; no. 325.
Temas:
Ricci flow.
Geometry, Riemannian.
Mathematics.
Flot de Ricci.
Géométrie de Riemann.
Mathématiques.
MATHEMATICS > Geometry > Differential.
Ricci flow
Instructional and educational works.
Matériel d'éducation et de formation.
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 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
  • 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
  • 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
  • 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
  • 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.

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