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Springer Monographs in Mathematics For other titles published in this series, go to http://www.springer.com/series/3733 Olli Martio • Vladimir Ryazanov • Uri Srebro • Eduard Yakubov Moduli in Modern Mapping Theory With 12 Illustrations 123 Olli Martio Vladimir Ryazanov University of Helsinki Institute of Applied Mathematics Helsinki and Mechanics of National Academy Finland of Sciences of Ukraine olli.martio@helsinki.fi Donetsk Ukraine [email protected] Uri Srebro Eduard Yakubov Technion - Israel Institute H.I.T. - Holon Institute of Technology of Technology Holon Haifa Israel Israel [email protected] [email protected] ISSN: 1439-7382 ISBN: 978-0-387-85586-8 e-ISBN: 978-0-387-85588-2 DOI 10.1007/978-0-387-85588-2 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 connec- tion 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 map- pings 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 geomet- ric 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 O. Martio Donetsk V. Ryazanov Haifa U. Srebro Holon E. Yakubov 2007 Contents 1 Introduction and Notation ...................................... 1 2 Moduli and Capacity ........................................... 7 2.1 Introduction . ............................................ 7 2.2 Moduli in Metric Spaces. ................................. 7 2.3 Conformal Modulus ........................................ 11 2.4 Geometric Definition for Quasiconformality . ............... 13 2.5 Modulus Estimates . ........................................ 14 2.6 Upper Gradients and ACCp Functions . ...................... 17 n 2.7 ACCp Functions in R and Capacity . ...................... 21 2.8 Linear Dilatation . ........................................ 25 2.9 Analytic Definition for Quasiconformality . ............... 31 2.10 Rn as a Loewner Space . ................................. 34 2.11Quasisymmetry............................................ 40 3 Moduli and Domains ........................................... 47 3.1 Introduction . ............................................ 47 3.2 QED Exceptional Sets . ................................. 48 3.3 QED Domains and Their Properties . ...................... 52 3.4 UniformandQuasicircleDomains............................ 55 3.5 Extension of Quasiconformal and Quasi-Isometric Maps . .... 62 3.6 Extension of Local Quasi-Isometries . ...................... 69 3.7 Quasicircle Domains and Conformal Mappings . ............... 71 3.8 On Weakly Flat and Strongly Accessible Boundaries . ........... 73 ∈ 1 4 Q-Homeomorphisms with Q Lloc ............................... 81 4.1 Introduction . ............................................ 81 4.2 Examples of Q-homeomorphisms............................. 82 n−1 4.3 Differentiability and KO(x, f ) ≤ Cn Q (x) a.e................. 83 1,1 4.4 Absolute Continuity on Lines and Wloc ........................ 86 4.5 LowerEstimateofDistortion................................. 89 x Contents 4.6 Removal of Singularities . ................................. 90 4.7 Boundary Behavior . ........................................ 91 4.8 Mapping Problems . ........................................ 92 5 Q-homeomorphisms with Q in BMO ............................. 93 5.1 Introduction . ............................................ 93 5.2 MainLemmaonBMO...................................... 94 5.3 Upper Estimate of Distortion ................................. 96 5.4 Removal of Isolated Singularities ............................. 97 5.5 On Boundary Correspondence . ............................. 97 5.6 Mapping Problems . ........................................ 99 5.7 SomeExamples............................................101 6 More General Q-Homeomorphisms ..............................103 6.1 Introduction . ............................................103 6.2 Lemma on Finite Mean Oscillation . ......................104 6.3 On Super Q-Homeomorphisms...............................108 6.4 Removal of Isolated Singularities .............................109 6.5 Topological Lemmas . .................................114 6.6 On Singular Sets of Length Zero ..............................118 6.7 Main Lemma on Extension to Boundary . ......................121 6.8 Consequences for Quasiextremal Distance Domains . ...........123 6.9 On Singular Null Sets for Extremal Distances ...................125 6.10 Applications to Mappings in Sobolev Classes . ...............126 7RingQ-Homeomorphisms .......................................131 7.1 Introduction . ............................................131 7.2 On Normal Families of Maps in Metric Spaces . ...............132 7.3 Characterization of Ring Q-Homeomorphisms..................135 7.4 EstimatesofDistortion......................................137 7.5 On Normal Families of Ring Q-Homeomorphisms...............141 7.6 On Strong Ring Q-Homeomorphisms..........................142 8 Mappings with Finite Length Distortion (FLD) ....................145 8.1 Introduction . ............................................145 8.2 Moduli of Cuttings and Extensive Moduli ......................147 8.3 FMD Mappings ............................................149 8.4 FLD Mappings ............................................152 8.5 Uniqueness Theorem . .................................154 8.6 FLD and Q-Mappings . .................................156 8.7 OnFLDHomeomorphisms..................................159 8.8 OnSemicontinuityofOuterDilatations........................164 8.9 On Convergence of Matrix Dilatations . ......................169 8.10 Examples and Subclasses . .................................172 Contents xi 9LowerQ-Homeomorphisms .....................................175 9.1 Introduction . ............................................175 9.2 On Moduli of Families of Surfaces . ......................176 9.3 Characterization of Lower Q-Homeomorphisms.................180 9.4 EstimatesofDistortion......................................183 9.5 Removal of Isolated Singularities .............................184 9.6 On Continuous Extension to Boundary Points ...................185 9.7 OnOneCorollaryforQEDDomains..........................186 9.8 On Singular Null Sets for Extremal Distances ...................186 9.9 LemmaonClusterSets......................................187 9.10 On Homeomorphic Extensions to Boundaries . ...............190 10 Mappings with Finite Area Distortion ............................193 10.1 Introduction . ............................................193 10.2 Upper Estimates of Moduli . .................................194 10.3 On Lower Estimates of Moduli . .............................198 10.4 Removal of isolated singularities ..............................199 10.5 Extension to Boundaries . .................................200 10.6 Finitely Bi-Lipschitz Mappings . .............................202 11 On Ring Solutions of the Beltrami Equation .......................205 11.1 Introduction . ............................................205 11.2 Finite Mean Oscillation . .................................207 11.3 Ring Q-HomeomorphismsinthePlane ........................211 11.4DistortionEstimates........................................216 11.5 General Existence Lemma and Its Corollaries . ...............224 11.6 Representation, Factorization and Uniqueness Theorems . ....228 11.7Examples.................................................232 12 Homeomorphisms with Finite Mean Dilatations ...................237 12.1 Introduction . ............................................237 12.2 Mean Inner and Outer Dilatations .............................239 12.3 On Distortion of p-Moduli . .................................242 12.4 Moduli of Surface Families Dominated by Set Functions . ....244 12.5 Alternate Characterizations of Classical Mappings . ...........247 12.6 Mappings (α,β)-Quasiconformal in the Mean . ...............249 12.7 Coefficients of Quasiconformality of Ring Domains . ...........251 13 On Mapping Theory in Metric Spaces ............................257 13.1 Introduction . ............................................257 13.2 Connectedness in Topological Spaces . ......................259 13.3 On Weakly Flat and Strongly Accessible
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