Moduli in Modern Mapping Theory
The purpose of this book is to present a modern account of mapping theory with emphasis on quasiconformal mapping and its generalizations. The modulus method was initiated by Arne Beurling and Lars Ahlfors to study conformal mappings, and later this metho
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Olli Martio Vladimir Ryazanov Uri Srebro Eduard Yakubov •
Moduli in Modern Mapping Theory With 12 Illustrations
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Olli Martio University of Helsinki Helsinki Finland [email protected]
Vladimir Ryazanov Institute of Applied Mathematics and Mechanics of National Academy of Sciences of Ukraine Donetsk Ukraine [email protected]
Uri Srebro Technion - Israel Institute of Technology Haifa Israel [email protected]
Eduard Yakubov H.I.T. - Holon Institute of Technology Holon Israel [email protected]
ISSN: 1439-7382 ISBN: 978-0-387-85586-8 DOI 10.1007/978-0-387-85588-2
e-ISBN: 978-0-387-85588-2
Library of Congress Control Number: 2008939873 c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com
Dedicated to 100 Years of Lars Ahlfors
Preface
The purpose of this book is to present modern developments and applications of the techniques of modulus or extremal length of path families in the study of mappings in Rn , n ≥ 2, and in metric spaces. The modulus method was initiated by Lars Ahlfors and Arne Beurling to study conformal mappings. Later this method was extended and enhanced by several other authors. The techniques are geometric and have turned out to be an indispensable tool in the study of quasiconformal and quasiregular mappings as well as their generalizations. The book is based on rather recent research papers and extends the modulus method beyond the classical applications of the modulus techniques presented in many monographs.
Helsinki Donetsk Haifa Holon 2007
O. Martio V. Ryazanov U. Srebro E. Yakubov
Contents
1
Introduction and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2
Moduli and Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Moduli in Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Conformal Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Geometric Definition for Quasiconformality . . . . . . . . . . . . . . . . . . . . 2.5 Modulus Estimate
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