EMS Tracts in Mathematics 19 EMS Tracts in Mathematics Editorial Board: Carlos E. Kenig (The University of Chicago, USA) Andrew Ranicki (The University of Edinburgh, Great Britain) Michael Röckner (Universität Bielefeld, Germany, and Purdue University, USA) Vladimir Turaev (Indiana University, Bloomington, USA) Alexander Varchenko (The University of North Carolina at Chapel Hill, USA) This series includes advanced texts and monographs covering all fields in pure and applied mathematics. Tractswill give a reliable introduction and reference to special fields of current research. The books in the series will in most cases be authored monographs, although edited volumes may be published if appropriate. They are addressed to graduate students seeking access to research topics as well as to the experts in the field working at the frontier of research. 1 Panagiota Daskalopoulos and Carlos E. Kenig, Degenerate Diffusions 2 Karl H. Hofmann and Sidney A. Morris, The Lie Theory of Connected Pro-Lie Groups 3 Ralf Meyer, Local and Analytic Cyclic Homology 4 Gohar Harutyunyan and B.-Wolfgang Schulze, Elliptic Mixed, Transmission and Singular Crack Problems 5 Gennadiy Feldman, Functional Equations and Characterization Problems on Locally Compact Abelian Groups , 6 Erich Novak and Henryk Wozniakowski, Tractability of Multivariate Problems. Volume I: Linear Information 7 Hans Triebel, Function Spaces and Wavelets on Domains 8 Sergio Albeverio et al., The Statistical Mechanics of Quantum Lattice Systems 9 Gebhard Böckle and Richard Pink,Cohomological Theory of Crystals over Function Fields 10 Vladimir Turaev,Homotopy Quantum Field Theory 11 Hans Triebel,Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration , 12 Erich Novak and Henryk Wozniakowski, Tractability of Multivariate Problems. Volume II: Standard Information for Functionals 13 Laurent Bessières et al.,Geometrisation of 3-Manifolds 14 Steffen Börm,Efficient Numerical Methods for Non-local Operators. (cid:1)2-Matrix Compression, Algorithms and Analysis 15 Ronald Brown, Philip J. Higgins and Rafael Sivera,Nonabelian Algebraic Topology. Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids 16 Marek Janicki and Peter Pflug,Separately Analytical Functions 17 Anders Björn and Jana Björn,Nonlinear Potential Theory on Metric Spaces , 18 Erich Novak and Henryk Wozniakowski, Tractability of Multivariate Problems. Volume III: Standard Information for Operators Bogdan Bojarski Vladimir Gutlyanskii Olli Martio Vladimir Ryazanov Infinitesimal Geometry of Quasiconformal and Bi-Lipschitz Mappings in the Plane Authors: Bogdan Bojarski Vladimir Gutlyanskii Institute of Mathematics of the Department of Partial Differential Equations Polish Academy of Sciences Institute of Applied Mathematics and Mechanics Warsaw, Poland National Academy of Sciences of Ukraine Donetsk, Ukraine E-mail: [email protected] E-mail: [email protected] Olli Martio Vladimir Ryazanov University of Helsinki Department of Function Theory Finnish Academy of Science and Letters Institute of Applied Mathematics and Mechanics Helsinki, Finland National Academy of Sciences of Ukraine Donetsk, Ukraine E-mail: [email protected] E-mail: [email protected] 2010 Mathematical Subject Classification: 30C65, 30C75, 35J46, 35J50, 35J56, 35J70, 35Q35, 35Q60, 37F30, 37F40, 37F45, 57R99 Key words: quasiconformal mappings, bi-Lipschitz mappings, Beltrami equations, local and boundary behavior, infinitesimal space, convergence and compactness theory, asymptotic linearity, rotation pro- blems, conformal differentiability ISBN 978-3-03719-122-4 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadca- sting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © European Mathematical Society 2013 Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the authors’ TEX files: I. Zimmermann, Freiburg Printed in Germany 9 8 7 6 5 4 3 2 1 Preface Thepurposeofthisbookistopresentrecentadvancesinthetheoryoflocalpropertiesof mappingsintheplane. Themainemphasisisinthealmosteverywheredifferentiable homeomorphic mappings. Thus quasiconformal and bi-Lipschitz mappings and the methodstostudythesemappingsformthecoreofthebook. QuasiconformalmappingshaveturnedouttobeinstrumentalinthestudyofRie- mann surfaces, Teichmüller spaces, Kleinian groups, meromorphic functions, holo- morphic motion complex dynamics, Clifford analysis and tomography. Bi-Lipschitz mappingsareusedforexampleintheelasticityandcontroltheoryandthereareunex- pectedconnectionsbetweenthesemappingclasses. The book consists of three parts. The first part contains some problems from analysisandmathematicalphysics,thestudyofwhichleadsnaturallytotheBeltrami equationandthereforetothequasiconformalmappings. Thispartismainlyintroduc- toryandintendedtoreadersnotfamiliarwithquasiconformalmappings. Muchofthe materialcanbefoundinotherbooksandhenceseveralproofsareomitted. However, inChapter5.1wepresentseveralexamplesofquasiconformalmappingswhichexhibit thecomplicatedlocalbehaviorofquasiconformalmappings. PartIIisintendedforresearchersinterestedinnewaspectsofinfinitesimalbehavior ofmappings. Thecompactnesspropertiesofquasiconformalmappingsmakeitpossible tostudytheinfinitesimalbehaviorofaquasiconformalmappingatapointwherethe mappingisnotdifferentiable. Thisleadstotheconceptofaninfinitesimalspaceand the concept is used in subsequent chapters to study local properties of mappings. At theendofpartIIweconsiderclassicallocalregularityresultsontheboundaryfroma newpointofview. InPartIIIweapplythequasiconformalfunctiontheorytostudyanon-linearelas- ticityproblemandbi-Lipschitzmappings. Newmethodsareusedtostudyinteriorand boundaryvariationofquasiconformalmappingsandcriteriaofunivalence. Throughoutthebookwehavetriedtoillustratetheresultsbyexamples. Manyof themhavenotbeenpublishedinmonographsbefore. Thisbookisaddressedtotheexpertsinmoderngeometricanalysis,quasiconformal mappings and extensions, non-linear elasticity theory as well as to the beginning re- searchersandgraduatestudentswithayear’sbackgroundincomplexvariablesseeking accesstoresearchtopics. WearegratefultoAnatolyGolberg,SamuelKrushkal,MattiVuorinenandanony- mousrefereesformanyusefulcommentsgivenondraftversionsofthisbook. Wealsoappreciatehelpfulandhighlyprofessionalassistancefromthestaffofthe European Mathematical Society Publishing House and wish to thank Irene Zimmer- mannforhercarefulreadingofthemanuscriptandmanyusefulcomments. Warsaw–Donetsk–Helsinki2013 B.Bojarski,V.Gutlyanski˘ı, O.Martio,V.Ryazanov Contents Preface v I QuasiconformalMappingsinthePlane 1 1 Backgroundofthetheory 3 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Quasiconformaldiffeomorphisms . . . . . . . . . . . . . . . . . . 6 1.3 Grötzsch’smappingproblem . . . . . . . . . . . . . . . . . . . . . 7 1.4 Conformalstructuresonsurfaces . . . . . . . . . . . . . . . . . . . 10 1.5 Conductivityininhomogeneousmedia . . . . . . . . . . . . . . . . 12 1.6 Holomorphicmotions . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.7 Streamliningofaballinspace . . . . . . . . . . . . . . . . . . . . 15 1.8 Bi-Lipschitzrotation . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Conformalinvariants 19 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Extremallength . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Modulusofacurvefamily . . . . . . . . . . . . . . . . . . . . . . 21 2.4 Modulusofacircularannulus . . . . . . . . . . . . . . . . . . . . 25 2.5 Modulusofarectangle . . . . . . . . . . . . . . . . . . . . . . . . 25 2.6 Modulusofquadrilateralsandringdomains . . . . . . . . . . . . . 26 2.7 Grötzsch’sandTeichmüller’smodulustheorem . . . . . . . . . . . 28 2.8 Diameterestimates . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Definitionsofquasiconformalmaps 34 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2 Geometricdefinition . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3 Absolutecontinuityonlines . . . . . . . . . . . . . . . . . . . . . 35 3.4 Differentiabilityalmosteverywhere . . . . . . . . . . . . . . . . . 35 3.5 Dilatationcondition . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.6 Integrabilitycondition . . . . . . . . . . . . . . . . . . . . . . . . 37 3.7 Pointwisedilatation . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.8 Astala’sregularitytheorem . . . . . . . . . . . . . . . . . . . . . . 39 3.9 Reich–Walczak’stypeintegralmodulusestimates . . . . . . . . . . 41 3.10 Analyticdefinition . . . . . . . . . . . . . . . . . . . . . . . . . . 46 viii Contents 4 Compactnessandconvergencetheory 49 4.1 Generalconvergenceproperties . . . . . . . . . . . . . . . . . . . . 49 4.2 Equicontinuityproperties . . . . . . . . . . . . . . . . . . . . . . . 53 4.3 Dilatationandconvergence . . . . . . . . . . . . . . . . . . . . . . 56 5 Beltramidifferentialequation 58 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.2 CauchyandHilberttransformations . . . . . . . . . . . . . . . . . 59 5.3 Tricomi’stypeintegralequations . . . . . . . . . . . . . . . . . . . 62 5.4 Existenceandrepresentationtheorems . . . . . . . . . . . . . . . . 65 5.5 MeasurableRiemannMappingTheorem . . . . . . . . . . . . . . . 67 5.6 Higherintegrabilityexponent . . . . . . . . . . . . . . . . . . . . . 69 5.7 Dependenceonparameter. . . . . . . . . . . . . . . . . . . . . . . 70 5.8 ProofforAstala’stheorem . . . . . . . . . . . . . . . . . . . . . . 74 5.9 Examplesofquasiconformalmappings . . . . . . . . . . . . . . . . 77 5.10 Examplesofquasiconformalmappings . . . . . . . . . . . . . . . . 77 II InfinitesimalGeometryofQuasiconformalMaps 85 6 Infinitesimalspace 87 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.2 Convergencetheoremsandmajorizingmetrics . . . . . . . . . . . . 89 6.3 Definitionoftheinfinitesimalspace . . . . . . . . . . . . . . . . . 91 6.4 SimpleT.z ;f/ . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 0 6.5 Approximatecontinuity . . . . . . . . . . . . . . . . . . . . . . . . 98 6.6 Weakconformalityproperty . . . . . . . . . . . . . . . . . . . . . 99 6.7 Asymptoticsymmetry . . . . . . . . . . . . . . . . . . . . . . . . . 100 7 Asymptoticallyconformalcurves 107 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.2 Definitionofasymptoticallyconformalcurves . . . . . . . . . . . . 107 7.3 Asymptotichomogeneityandconformality. . . . . . . . . . . . . . 108 7.4 Criteriaforasymptoticconformality . . . . . . . . . . . . . . . . . 112 7.5 Asymptoticsymmetryandconformality . . . . . . . . . . . . . . . 113 8 Conformaldifferentiability 117 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 8.2 Teichmüller–Wittich–Belinski˘ı’stypetheorem . . . . . . . . . . . . 119 8.3 Modulusestimates . . . . . . . . . . . . . . . . . . . . . . . . . . 120 8.4 Rotationtheorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.5 ProofforConformalityTheorems . . . . . . . . . . . . . . . . . . 126 8.6 Conformalityonaset . . . . . . . . . . . . . . . . . . . . . . . . . 129 Contents ix 9 Pointsofmaximalstretching 131 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 9.2 Sufficientconditions . . . . . . . . . . . . . . . . . . . . . . . . . 132 9.3 Maximalstretchingcriterion . . . . . . . . . . . . . . . . . . . . . 135 10 Lipschitzcontinuityofquasiconformalmaps 138 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 10.2 Growthestimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 10.3 ConditionsforLipschitzcontinuity . . . . . . . . . . . . . . . . . . 141 10.4 WeakLipschitzcontinuity . . . . . . . . . . . . . . . . . . . . . . 145 11 Regularityofquasiconformalcurves 150 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 11.2 Regularityofboundarycorrespondence . . . . . . . . . . . . . . . 150 11.3 Smoothnessofquasicircles . . . . . . . . . . . . . . . . . . . . . . 154 12 Regularityofconformalmapsattheboundary 157 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 12.2 Lindelöf’ssmoothnesstheorem . . . . . . . . . . . . . . . . . . . . 157 12.3 Warschawski’sConformalityTheorem . . . . . . . . . . . . . . . . 159 III ApplicationsofQuasiconformalMaps 161 13 John’srotationproblem 163 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 13.2 Mainresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 13.3 ProofofJohn’srotationproblem . . . . . . . . . . . . . . . . . . . 165 13.4 Factoringofspiral-likemaps . . . . . . . . . . . . . . . . . . . . . 168 14 Variationofquasiconformalmaps 172 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 14.2 Variationalprocedure . . . . . . . . . . . . . . . . . . . . . . . . . 172 14.3 Necessaryconditionsforextremum . . . . . . . . . . . . . . . . . 173 14.4 Linearpartialdifferentialsystems . . . . . . . . . . . . . . . . . . 175 15 Criteriaofunivalence 180 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 15.2 Ahlfors–Weill’scriteria . . . . . . . . . . . . . . . . . . . . . . . . 181 Bibliography 185 Index 203