Harmonic Maps Between Surfaces (with a Special Chapter on Conformal Mappings)
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JUrgen Jost
Harmonic Maps Between Surfaces (with a Special Chapter on Conformal Mappings)
Springer-Verlag Berlin Heidelberg New York Tokyo 1984
Author
JUrgen Jost Mathematisches Institut der Universitat Wegelerstr. 10, 5300 Bonn, Federal Republic of Germany
AMS Subject Classification (1980): 58E20; 30C70, 32G15, 35J60 ISBN 3-540-13339-9 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13339-9 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
Dedicated to the memory of Dieter Kieven
PRE F ACE
The purpose of these Lecture Notes is twofold. On one hand, I want togive a fairly complete and self contained account of the results on harmonic maps between surfaces. On the other hand, these notes should also serve as an introduction to the theory of harmonic maps in general; therefore, whenever appropriate, I point out which of the twodimensional results pertain to higher dimensions and which do not, and I try to give some references and an idea of the respective proof. For a more complete account in this direction, however, the reader should consult the several excellent survey articles of Eells and Lemaire. An essential aim of this book is to show the variety of methods and
the interplay of different fields in the theory of harmonic maps, in particular the calculus of variations, partial differential equations, differential geometry, algebraic topology, and complex analysis. Thus, the concept of this book is strongly opposed to the view of a mere specialist. In particular, I think that a completely unified treatment of the topic is neither possible nor desirable. Nevertheless, I believe that this treatment contains several simplifications and unifications compared to the presentations available in the existing literature. This book is not intended as a mere enumeration of unrelated results. On the contrary, the sequence of the chapters also reflects a logical order, and many different tools have to be constructed, until the results of the three final chapters can be proved. In particular, conformal mappings are used in a much more thorough way than in the existing literature. An outline of the contents now follows. After giving an account of the history and presenting the definition of harmonic maps from several points of view in chapter 1, we start in chapter 2 with some geometric considerations. These concern convex discs on surfaces, and the result is roughly that if on a disc there are no conjugate points then there are also n
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