Lectures on the Ricci flow [E-Book] / Peter Topping.
Topping, Peter, (author)
Cambridge : Cambridge University Press, 2006
1 online resource (x, 113 pages)
London Mathematical Society lecture note series ; 325
Full Text
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505 2 0 |g Preface --  |g Introduction --  |t Riemannian geometry background --  |t The maximum princople --  |t Comments on existence theory for parabolic PDE --  |t Existence theory for the Ricci flow --  |t Ricci flow as a gradient flow --  |t Compactness of Riemannian manifolds and flows --  |t Perelman's w entropy functional --  |t Curvature pinching and preserved curvature properties under Ricci flow --  |t Three-manifolds with positive Ricci curvature, and beyond. 
520 |a 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 whichcarries a metric of positive Ricci curvature is a spherical space form. 
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