The Ricci Flow in Riemannian Geometry A Complete Proof of the Differ
This book focuses on Hamilton's Ricci flow, beginning with a detailed discussion of the required aspects of differential geometry, progressing through existence and regularity theory, compactness theorems for Riemannian manifolds, and Perelman's noncollap
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2011
Ben Andrews · Christopher Hopper
The Ricci Flow in Riemannian Geometry A Complete Proof of the Differentiable 1/4-Pinching Sphere Theorem
123
Ben Andrews
Christopher Hopper
Australian National University Mathematical Sciences Institute ACT 0200 Australia [email protected]
University of Oxford Mathematical Institute St Giles’ 24-29 OX1 3LB Oxford United Kingdom [email protected]
ISBN: 978-3-642-16285-5 e-ISBN: 978-3-642-16286-2 DOI: 10.1007/978-3-642-16286-2 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Mathematics Subject Classification (2010): 35-XX, 53-XX, 58-XX © Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
For in the very torrent, tempest, and as I may say, whirlwind of your passion, you must acquire and beget a temperance that may give it smoothness. — Shakespeare, Hamlet.
Preface
There is a famous theorem by Rauch, Klingenberg and Berger which states that a complete simply connected n-dimensional Riemannian manifold, for which the sectional curvatures are strictly between 1 and 4, is homeomorphic to a n-sphere. It has been a longstanding open conjecture as to whether or not the ‘homeomorphism’ conclusion could be strengthened to a ‘diffeomorphism’. Since the introduction of the Ricci flow by Hamilton [Ham82b] some two decades ago, there have been several inroads into this problem – particularly in dimensions three and four – which have thrown light upon a possible proof of this result. Only recently has this conjecture (and a considerably stronger generalisation) been proved by Simon Brendle and Richard Schoen. The aim of the present book is to provide a unified expository account of the differentiable 1/4-pinching sphere theorem together with the necessary background material and recent convergence theory for the Ricci flow in n-dimensions. This account should be accessible to anyone familiar with enough differential geometry to feel comfortable with tensors, covariant derivatives, and normal coordinates; and enough analysis to follow s
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