Lectures on the Ricci FlowCambridge University Press, 2006/10/12 - 113 ページ 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. |
目次
Introduction | 1 |
Riemannian geometry background | 16 |
The maximum principle | 35 |
Comments on existence theory for parabolic PDE | 43 |
Ricci flow as a gradient flow | 55 |
Compactness of Riemannian manifolds and flows | 65 |
Perelmans W entropy functional | 71 |
Curvature pinching and preserved curvature properties | 88 |
Threemanifolds with positive Ricci curvature and beyond | 105 |
多く使われている語句
algebraic apply arbitrary assume Bianchi identity bound bundle calculate Chapter closed manifold Combining compact compute condition connection consider constant convergence convex Corollary defined definition denote depending derivative diffeomorphisms differential dimensions discussed eigenvalues equation equivalently estimates evolution evolve example exists expression extended fact field final finite formula function geometry give given gradient groups heat hence identity independent inequality interval invariant Lemma limit lower metric metric g notation Note operator parabolic parallel particular pinch positive preserved principle Proof Proposition prove Remark rescaling respect result Ric(X Ricci curvature Ricci flow Riemannian satisfying scalar curvature sectional curvature sense sequence simply singularities smooth soliton solution space Suppose symmetric tensor Theorem theory vector volume weakly write