Lectures on the Ricci Flow

Peter Topping0521689473, 9780521689472

Hamilton’s Ricci flow has attracted considerable attention since its introduction in 1982, owing partly to its promise in addressing the Poincaré conjecture and Thurston’s geometrization conjecture. This book gives a concise introduction to the subject with the hindsight of Perelman’s breakthroughs from 2002/2003. After describing the basic properties of, and intuition behind the Ricci flow, core elements of the theory are discussed such as consequences of various forms of maximum principle, issues related to existence theory, and basic properties of singularities in the flow. A detailed exposition of Perelman’s entropy functionals is combined with a description of Cheeger-Gromov-Hamilton compactness of manifolds and flows to show how a ‘tangent’ flow can be extracted from a singular Ricci flow. Finally, all these threads are pulled together to give a modern proof of Hamilton’s theorem that a closed three-dimensional manifold which carries a metric of positive Ricci curvature is a spherical space form.

---

Table of contents :
Ricci flow: what is it, and from where did it come?……Page 7
Ricci solitons……Page 9
Parabolic rescaling of Ricci flows……Page 12
Two dimensions……Page 13
Three dimensions……Page 14
The topology and geometry of manifolds in low dimensions……Page 18
Using Ricci flow to prove topological and geometric results……Page 22
Notation and conventions……Page 25
The formulae……Page 29
The calculations……Page 33
Laplacian of the curvature tensor……Page 40
Evolution of curvature and geometric quantities under Ricci flow……Page 42
Statement of the maximum principle……Page 45
Basic control on the evolution of curvature……Page 46
Global curvature derivative estimates……Page 50
Linear scalar PDE……Page 54
The principal symbol……Page 55
Generalisation to Vector Bundles……Page 57
Properties of parabolic equations……Page 59
Ricci flow is not parabolic……Page 60
Short-time existence and uniqueness: The DeTurck trick……Page 61
Curvature blow-up at finite-time singularities……Page 64
Gradient of total scalar curvature and related functionals……Page 68
The F-functional……Page 69
A gradient flow formulation……Page 71
The classical entropy……Page 75
The zeroth eigenvalue……Page 77
Compactness of Riemannian manifolds and flows……Page 79
Convergence and compactness of manifolds……Page 80
Convergence and compactness of flows……Page 83
Blowing up at singularities I……Page 84
Definition, motivation and basic properties……Page 86
Monotonicity of W……Page 92
No local volume collapse where curvature is controlled……Page 95
Volume ratio bounds imply injectivity radius bounds……Page 101
Blowing up at singularities II……Page 103
Overview……Page 105
The Einstein Tensor, E……Page 106
Evolution of E under the Ricci flow……Page 107
The Uhlenbeck Trick……Page 108
Formulae for parallel functions on vector bundles……Page 110
An ODE-PDE theorem……Page 113
Applications of the ODE-PDE theorem……Page 116
Hamilton’s theorem……Page 124
Beyond the case of positive Ricci curvature……Page 126
Connected sum……Page 128
Index……Page 130