Boundary Behaviour of Conformal Maps
We study the boundary behaviour of a conformal map of the unit disk onto an arbitrary simply connected plane domain. A principal aim of the theory is to obtain a one-to-one correspondence between analytic properties of the function and geometrie propertie
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Editors M. Artin S. S. Chern 1. Coates 1. M. Fröhlich H. Hironaka F. Hirzebruch L. Hörmander C. C. Moore 1. K. Moser M. Nagata W. Schmidt D. S. Soott Ya. G. Sinai 1. Tits M. Waldschmidt S.Watanabe
Managing Editors M. Berger B. Eckmann S. R. S. Varadhan
eh. Pommerenke
Boundary Behaviour of Conformal Maps With 76 Figures
Springer-Verlag Berlin Heidelberg GmbH
Christian Pommerenke Fachbereich Mathematik Technische Universität 1000 Berlin 12, FRG
Mathematics Subject Classification (1991):
30Cxx, 30D40, 30D45, 30D50, 28A78
ISBN 978-3-642-08129-3
Library of Congress Cataloging-in-Publication Data Pommerenke, Christian. Boundary behaviour of conformal maps ICh. Pommerenke. p. cm. - (Grundlehren der mathematischen Wissenschaften; 299) Includes bibliographical references and indexes. ISBN 978-3-642-08129-3 ISBN 978-3-662-02770-7 (eBook) DOI 10.1007/978-3-662-02770-7 1. Conformal mapping. 2. Boundary value problems. I. Title 11. Series. QA360.P66 1992 515'.9-dc20 92-10365 CIP This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concemed, specifically the rights oftranslation, reprinting, reuse ofillustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication ofthis publication or parts thereofis permitted only under the provisions ofthe German Copyright Law ofSeptember9, 1965, in its current version, and permission for use must a1ways be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations are liable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1992 Originally published by Springer-Verlag Berlin Heidelberg New York in 1992 Softcover reprint of the hardcover 1st edition 1992 Typesetting: Data conversion by Springer-Verlag 41/3140 - 5 4 3 21 0 Printed on acid-free paper
Preface
We study the boundary behaviour of a conformal map of the unit disk onto an arbitrary simply connected plane domain. A principal aim of the theory is to obtain a one-to-one correspondence between analytic properties of the function and geometrie properties of the domain. In the classical applications of conformal mapping, the domain is bounded by a piecewise smooth curve. In many recent applications however, the domain has a very bad boundary. It may have nowhere a tangent as is the case for Julia sets. Then the conformal map has many unexpected properties, for instance almost all the boundary is mapped onto almost nothing and vice versa. The book is meant for two groups of users. (1) Graduate students and others who, at various levels, want to learn about conformal mapping. Most sections contain exercises to test the understanding. They tend to be fairly simple and only a few contain new material. Prerequisites are general real and complex analyis including the basic facts about conformal mapping (e.g. AhI66a). (2) Non-experts who want to get an idea of a particular aspect of conformal mapping in order to find something useful for their work. Most chapters therefore begin wi
