Riemannian Geometry and Geometric Analysis
This established reference work continues to lead its readers to some of the hottest topics of contemporary mathematical research.This new edition introduces and explains the ideas of the parabolic methods that have recently found such a spectacular succe
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Riemannian Geometry and Geometric Analysis Fifth Edition
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J¨urgen Jost Max Planck Institute for Mathematics in the Sciences Inselstr. 22 04103 Leipzig Germany [email protected]
ISBN 978-3-540-77340-5
e-ISBN 978-3-540-77341-2
Universitext Library of Congress Control Number: 2008921366 Mathematics Subject Classification (2000): 53B21, 53L20, 32C17, 35I60, 49-XX, 58E20, 57R15 c Springer-Verlag Berlin Heidelberg 1995, 1998, 2001, 2005, 2008 ° 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: design & production GmbH, Heidelberg Printed on acid-free paper 987654321 springer.com
Dedicated to Shing-Tung Yau, for so many discussions about mathematics and Chinese culture
Preface
Riemannian geometry is characterized, and research is oriented towards and shaped by concepts (geodesics, connections, curvature, . . . ) and objectives, in particular to understand certain classes of (compact) Riemannian manifolds defined by curvature conditions (constant or positive or negative curvature, . . . ). By way of contrast, geometric analysis is a perhaps somewhat less systematic collection of techniques, for solving extremal problems naturally arising in geometry and for investigating and characterizing their solutions. It turns out that the two fields complement each other very well; geometric analysis offers tools for solving difficult problems in geometry, and Riemannian geometry stimulates progress in geometric analysis by setting ambitious goals. It is the aim of this book to be a systematic and comprehensive introduction to Riemannian geometry and a representative introduction to the methods of geometric analysis. It attempts a synthesis of geometric and analytic methods in the study of Riemannian manifolds. The present work is the fifth edition of my textbook on Riemannian geometry and geometric analysis. It has developed on the basis of several graduate courses I taught at the Ruhr-University Bochum and the University of Leipzig. The main new features of the present edition are the systematic inclusion of flow equations and a mathematical treatment of the nonlinear sigma model of quantum field theory. These new topics also led to a systematic reorganization of the other material. Naturally, I have also included several small
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