Moduli of Families of Curves for Conformal and Quasiconformal Mappings
The monograph is concerned with the modulus of families of curves on Riemann surfaces and its applications to extremal problems for conformal, quasiconformal mappings, and the extension of the modulus onto Teichmüller spaces. The main part of the monograp
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3 Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Tokyo
Alexander Vasil’ev
Moduli of Families of Curves for Conformal and Quasiconformal Mappings
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Author Alexander Vasil’ev Departamento de Matem´atica Universidad T´ecnica Federico Santa Mar´ıa Casilla 110-V, Valpara´ıso, Chile E-mail: [email protected]
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Vasil'ev, Aleksandr: Moduli of families of curves for conformal and quasiconformal mappings / Alexander Vasil'ev. - Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 2002 (Lecture notes in mathematics ; 1788) ISBN 3-540-43846-7
Mathematics Subject Classification (2000): 30C35, 30C55, 30C62, 30C75, 30F10, 30F60 ISSN 0075-8434 ISBN 3-540-43846-7 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm 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-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2002 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10878683
41/3142/du-543210 - Printed on acid-free paper
Preface
In the present monograph, we consider the extremal length method in its form of the method of moduli of families of curves in applications to the problems of conformal, quasiconformal mapping, and Teichm¨ uller spaces. This method going back to H. Gr¨ otzsch, A. Beurling, L. V. Ahlfors, J. Jenkins is now one of the basic methods in various parts of Analysis. Several surveys and monographs, e.g., [30], [64], [78], [107], [139] are devoted to the development of this method and applications. However, we want to give here a useful guide: how one can start to solve extremal problems of conformal mapping beginning with simple but famous classical theorems and ending at difficult new results. Some more non-traditional applications we consider in the quasiconformal case. The modulus method permits us to consider the problems in question from a single point of view. At the mid-century it was established that the classical methods of the geometric function theory could be extended to complex hyperbolic manifol
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