Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge. Band 26 A Series of Modem Surveys in Mathematics Editorial Board E. Bombieri, Princeton S. Feferman, Stanford M. Gromov, Bures-sur-Yvette H.W. Lenstra, Jr., Berkeley P.-L. Lions, Paris R. Remmert (Managing Editor), Munster W. Schmid, Cambridge, Mass. J-P. Serre, Paris J.Tits, Paris
Seppo Rickman Quasiregular Mappings Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Seppo Rickman Department of Mathematics University of Helsinki Hallituskatu 15 SF-On 1 00 Helsinki, Finland Mathematics Subject Classification (1991): 30C65, 30C75, 30D35, 32A22, 32H25, 32H30 With 5' Figures ISBN-13: 978-3-642-78203-9 e-isbn-13: 978-3-642-78201-5 DOl: 10.1007/978-3-642-78201-5 Library of Congress Cataloging-in-Publication Data. Rickman, S. Quasiregular mappings / Seppo Rickman. p. cm. - (Ergebnisse der Mathematik und ihrer Grenzgebiete; 3. Folge, Bd. 26) Includes bibliographical references and index. ISBN 3-540-56648-1 (Berlin: acid-free): DM 128.00. - ISBN 0-387-56648-1 (New York: acid-free) I. Quasiconformal mappings. I. Title. II. Series. QA360.R53 1993 SIS'.93-dc20 93-4824 CIP This work i$ 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 microfilms or in any other ways, 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 1993 Softcover reprint of the hardcover 1st edition 1993 Typesetting: Camera ready copy from the author using a Springer TEX macro package 41/3140-5432 10- Printed on acid-free paper
To Eini
Preface This monograph grew out from lecture series which I gave at the University of Helsinki in 1980, at the University of Minnesota in 1980 and 1983, and in Bonn at the Max-Planck-Institute in 1986. A great deal of the basic material originates from a seminar at the University of Helsinki during the academic year 1968-69 organized by OUi Martio, Jussi Viiisiilii, and myself. It was a suggestion of Frederick Gehring in the early 80's that I should publish the content of my courses on quasiregular mappings in book form. At that time the defect relation, which is one of the main results in the book, was not known in its sharp form. The proof of its sharpness has just recently been completed. This matter delayed considerably the finishing of the text. During the preparation I have received much help from many people. I acknowledge my sincerest gratitude to Bruce Palka, who read most of the text and made numerous suggestions for improvements, especially to the language. I am also greatly indebted to Matti Vuorinen for careful reading of the manuscript and for valuable comments. During various stages of the work I have had useful discussions with Juha Heinonen, Ilkka Holopainen, Tadeusz Iwaniec, Tero Kilpeliiinen, Peter Lindqvist, OUi Martio, Pierre Pansu, and Jussi ViiisiiHi. To all of them I would like to express my gratitude. Particular thanks are due to Jouni Kankaanpiiii for writing the manuscript in 'lex with great care. He has also given valuable advice in the presentation of the text. I am grateful for the financial support from Suomen Kulttuurirahasto, the Academy of Finland, and the Finnish Society of Sciences and Letters. Finally, I thank the editors for accepting my text in Springer-Verlag's well-known series. Helsinki, Finland May, 1993 Seppo Rickman
Contents Introduction................... Chapter I. Basic Properties of Quasiregular Mappings l. ACU Mappings... 2. Quasiregular Mappings 3. Examples...... 4. Discrete Open Mappings Chapter II. Inequalities for Moduli of Path Families l. Modulus of a Path Family 2. The Ko-Inequality 3. Path Lifting... 4. Linear Dilatations 5. Poletskil's Lemma 6. Characterizations of Quasiregularity 7. Proof of Poletski'i's Lemma 8. Poletskil's Inequality 9. VaisaHi's Inequality 10. Capacity Inequalities Chapter III. Applications of Modulus Inequalities l. Global Distortion 2. Sets of Capacity Zero and Singularities 3. The Injectivity Radius of a Local Homeomorphism 4. Local Distortion..... 5. Bounds for the Local Index Chapter IV. Mappings into the n-sphere with Punctures l. Coverings Averages....... 2. The Analogue of Picard's Theorem 3. Mappings of a Ball.... Chapter V. Value Distribution l. Defect Relation 2. Coverings and Decomposition of Balls 1 4 4 10 13 16 24 24 30 32 35 39 42 45 49 51 53 59 59 64 68 71 73 78 78 81 89 94 94 97
X Contents 3. Estimates on Liftings........ 4. Extremal Maximal Sequences of Liftings 5. Effect of the Defect Sum on the Liftings 6. Completion of the Proof of Defect Relations 7. Mappings of the Plane 8. Order of Growth 9. Further Results Chapter VI. Variational Integrals and Quasiregular Mappings 1. Extremals of Variational Integrals 2. Extremals and Quasiregular Mappings 3. Growth Estimates for Extremals... 4. Differentiability of Quasiregular Mappings 5. Discreteness and Openness of Quasiregular Mappings 6. Pullbacks of General Kernels 7. Further Properties of Extremals 8. The Limit Theorem... Chapter VII. Boundary Behavior 1. Removability.... 2. Asymptotic and Radial Limits 3. Continuity Results and the Reflection Principle 4. The Wiener Condition.... 5. F-Harmonic Measure.... 6. Phragmen-Lindelof Type Theorems 7. Asymptotic Values Bibliography. List of Symbols Index... 101 105 108 111 116 120 124 129 130 135 137 142 145 148 152 157 162 162 168 174 180 184 187 193 198 208 210