Moduli in Modern Mapping Theory PDF
Preview Moduli in Modern Mapping Theory
Springer Monographs in Mathematics Forothertitlespublishedinthisseries,goto http://www.springer.com/series/3733 Olli Martio Vladimir Ryazanov Uri Srebro • • • Eduard Yakubov Moduli in Modern Mapping Theory With 12 Illustrations 123 OlliMartio VladimirRyazanov UniversityofHelsinki InstituteofAppliedMathematics Helsinki andMechanicsofNationalAcademy Finland ofSciencesofUkraine olli.martio@helsinki.fi Donetsk Ukraine [email protected] UriSrebro EduardYakubov Technion-IsraelInstitute H.I.T.-HolonInstituteofTechnology ofTechnology Holon Haifa Israel Israel [email protected] [email protected] ISSN:1439-7382 ISBN:978-0-387-85586-8 e-ISBN:978-0-387-85588-2 DOI10.1007/978-0-387-85588-2 LibraryofCongressControlNumber:2008939873 (cid:2)c SpringerScience+BusinessMedia,LLC2009 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,NY 10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnec- tionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orby similarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. Printedonacid-freepaper springer.com Dedicatedto100YearsofLarsAhlfors Preface The purpose of this book is to present modern developments and applications of thetechniquesofmodulusorextremallengthofpathfamiliesinthestudyofmap- 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- ricand have turned out tobe an indispensable tool in the study of quasiconformal and quasiregular mappings as well as their generalizations. The book is based on ratherrecentresearchpapersandextendsthemodulusmethodbeyondtheclassical applicationsofthemodulustechniquespresentedinmanymonographs. Helsinki O.Martio Donetsk V.Ryazanov Haifa U.Srebro Holon E.Yakubov 2007 Contents 1 IntroductionandNotation ...................................... 1 2 ModuliandCapacity ........................................... 7 2.1 Introduction ............................................... 7 2.2 ModuliinMetricSpaces..................................... 7 2.3 ConformalModulus ........................................ 11 2.4 GeometricDefinitionforQuasiconformality .................... 13 2.5 ModulusEstimates ......................................... 14 2.6 UpperGradientsandACCpFunctions ......................... 17 2.7 ACC FunctionsinRnandCapacity........................... 21 p 2.8 LinearDilatation ........................................... 25 2.9 AnalyticDefinitionforQuasiconformality...................... 31 2.10 RnasaLoewnerSpace...................................... 34 2.11 Quasisymmetry ............................................ 40 3 ModuliandDomains ........................................... 47 3.1 Introduction ............................................... 47 3.2 QEDExceptionalSets ...................................... 48 3.3 QEDDomainsandTheirProperties ........................... 52 3.4 UniformandQuasicircleDomains ............................ 55 3.5 ExtensionofQuasiconformalandQuasi-IsometricMaps.......... 62 3.6 ExtensionofLocalQuasi-Isometries .......................... 69 3.7 QuasicircleDomainsandConformalMappings ................. 71 3.8 OnWeaklyFlatandStronglyAccessibleBoundaries............. 73 4 Q-HomeomorphismswithQ∈L1 ............................... 81 loc 4.1 Introduction ............................................... 81 4.2 ExamplesofQ-homeomorphisms ............................. 82 4.3 Differentiabilityand K (x,f)≤C Qn−1(x) a.e. ................ 83 O n 4.4 AbsoluteContinuityonLinesandW1,1 ........................ 86 loc 4.5 LowerEstimateofDistortion................................. 89 x Contents 4.6 RemovalofSingularities .................................... 90 4.7 BoundaryBehavior ......................................... 91 4.8 MappingProblems ......................................... 92 5 Q-homeomorphismswithQinBMO ............................. 93 5.1 Introduction ............................................... 93 5.2 MainLemmaonBMO ...................................... 94 5.3 UpperEstimateofDistortion................................. 96 5.4 RemovalofIsolatedSingularities ............................. 97 5.5 OnBoundaryCorrespondence................................ 97 5.6 MappingProblems ......................................... 99 5.7 SomeExamples............................................101 6 MoreGeneralQ-Homeomorphisms ..............................103 6.1 Introduction ...............................................103 6.2 LemmaonFiniteMeanOscillation............................104 6.3 OnSuperQ-Homeomorphisms ...............................108 6.4 RemovalofIsolatedSingularities .............................109 6.5 TopologicalLemmas........................................114 6.6 OnSingularSetsofLengthZero..............................118 6.7 MainLemmaonExtensiontoBoundary .......................121 6.8 ConsequencesforQuasiextremalDistanceDomains .............123 6.9 OnSingularNullSetsforExtremalDistances...................125 6.10 ApplicationstoMappingsinSobolevClasses ...................126 7 RingQ-Homeomorphisms.......................................131 7.1 Introduction ...............................................131 7.2 OnNormalFamiliesofMapsinMetricSpaces..................132 7.3 CharacterizationofRingQ-Homeomorphisms ..................135 7.4 EstimatesofDistortion......................................137 7.5 OnNormalFamiliesofRingQ-Homeomorphisms...............141 7.6 OnStrongRingQ-Homeomorphisms..........................142 8 MappingswithFiniteLengthDistortion(FLD) ....................145 8.1 Introduction ...............................................145 8.2 ModuliofCuttingsandExtensiveModuli ......................147 8.3 FMDMappings ............................................149 8.4 FLDMappings ............................................152 8.5 UniquenessTheorem .......................................154 8.6 FLDandQ-Mappings.......................................156 8.7 OnFLDHomeomorphisms ..................................159 8.8 OnSemicontinuityofOuterDilatations ........................164 8.9 OnConvergenceofMatrixDilatations .........................169 8.10 ExamplesandSubclasses....................................172 Contents xi 9 LowerQ-Homeomorphisms .....................................175 9.1 Introduction ...............................................175 9.2 OnModuliofFamiliesofSurfaces ............................176 9.3 CharacterizationofLowerQ-Homeomorphisms.................180 9.4 EstimatesofDistortion......................................183 9.5 RemovalofIsolatedSingularities .............................184 9.6 OnContinuousExtensiontoBoundaryPoints...................185 9.7 OnOneCorollaryforQEDDomains ..........................186 9.8 OnSingularNullSetsforExtremalDistances...................186 9.9 LemmaonClusterSets......................................187 9.10 OnHomeomorphicExtensionstoBoundaries ...................190 10 MappingswithFiniteAreaDistortion ............................193 10.1 Introduction ...............................................193 10.2 UpperEstimatesofModuli ..................................194 10.3 OnLowerEstimatesofModuli ...............................198 10.4 Removalofisolatedsingularities..............................199 10.5 ExtensiontoBoundaries.....................................200 10.6 FinitelyBi-LipschitzMappings...............................202 11 OnRingSolutionsoftheBeltramiEquation.......................205 11.1 Introduction ...............................................205 11.2 FiniteMeanOscillation .....................................207 11.3 RingQ-HomeomorphismsinthePlane ........................211 11.4 DistortionEstimates ........................................216 11.5 GeneralExistenceLemmaandItsCorollaries...................224 11.6 Representation,FactorizationandUniquenessTheorems..........228 11.7 Examples .................................................232 12 HomeomorphismswithFiniteMeanDilatations ...................237 12.1 Introduction ...............................................237 12.2 MeanInnerandOuterDilatations .............................239 12.3 OnDistortionof p-Moduli...................................242 12.4 ModuliofSurfaceFamiliesDominatedbySetFunctions .........244 12.5 AlternateCharacterizationsofClassicalMappings...............247 12.6 Mappings(α,β)-QuasiconformalintheMean ..................249 12.7 CoefficientsofQuasiconformalityofRingDomains .............251 13 OnMappingTheoryinMetricSpaces ............................257 13.1 Introduction ...............................................257 13.2 ConnectednessinTopologicalSpaces..........................259 13.3 OnWeaklyFlatandStronglyAccessibleBoundaries.............262 13.4 OnFiniteMeanOscillationWithRespecttoMeasure ............263 13.5 OnContinuousExtensiontoBoundaries .......................267 13.6 OnExtendingInverseMappingstoBoundaries..................270 13.7 OnHomeomorphicExtensiontoBoundaries....................271 xii Contents 13.8 OnModuliofFamiliesofPathsPassingThroughPoint ..........272 13.9 OnWeaklyFlatSpaces .....................................274 13.10OnQuasiextremalDistanceDomains .........................277 13.11OnNullSetsforExtremalDistance...........................280 13.12OnContinuousExtensiontoIsolatedSingularPoints ............283 13.13OnConformalandQuasiconformalMappings ..................288 A ModuliTheory.................................................291 A.1 OnSomeResultsbyGehring.................................291 A.2 TheInequalitiesbyMartio–Rickman–Va¨isa¨la¨...................301 A.3 TheHesseEquality .........................................304 A.4 TheShlykEquality .........................................317 A.5 TheModulibyFuglede .....................................324 A.6 TheZiemerEquality........................................331 B BMOFunctionsbyJohn–Nirenberg..............................345 References.........................................................351 Index .............................................................365 Chapter 1 Introduction and Notation Mapping theory started in the 18th century. Beltrami, Caratheodory, Christoffel, Gauss,Hilbert,Liouville,Poincare´,Riemann,Schwarz,andsoonalllefttheirmarks inthistheory.Conformalmappingsandtheirapplicationstopotentialtheory,math- ematicalphysics,Riemannsurfaces,andtechnologyplayedakeyroleinthisdevel- opment. During the late 1920s and early 1930s, Gro¨tzsch, Lavrentiev, and Morrey in- troduced a more general and less rigid class of mappings that were later named quasiconformal.Verysoonquasiconformalmappingswereappliedtoclassicalprob- lemslikethecovering ofRiemann surfaces(Ahlfors),themoduliproblemofRie- mann surfaces (Teichmu¨ller), and the classification problem for simply connected Riemann surfaces (Volkovyski). Quasiconformal mappings were later defined in higher dimensions (Lavrentiev, Gehring, Va¨isa¨la¨) and were further extended to quasiregularmappings(Reshetnyak,Martio,Rickman,andVa¨isa¨la¨).Thequasireg- ularmappingsneednotbeinjectiveandinmanyaspectsaresimilartoanalyticfunc- tions.Themonographs[1,22,36,110,176,187,190,256,260,315,316,327–329]give acomprehensiveaccountoftheaforementionedtheoryanditsmorerecentachieve- ments. Recentlygeneralizationsofquasiconformalmappings,mappingsoffinitedistor- tion,havebeenstudiedintensively;see,e.g.,thepapers[19,45,46,54,79,111,115– 117,124,132,133,145,147–149,153–156,195,196,231–233,237,248–251] and themonograph[134].Quasisymmetryhasanaturalinterpretationinmetricspaces andquasiconformalityfromamoreanalyticpointofviewhasalsobeenstudiedin thesespaces;see,e.g.,[21,33,107,112,201,312].Thesetheoriescanbeappliedto mappings in the Carnot and Heisenberg groups; see, e.g., [108,109,166,167,197, 199,221,238,314,324–326]. The method of themodulus of apath family,or equivalently themethod ofex- tremal length, which was initiated by Ahlfors and Beurling in [5] for the study of conformal mapping, is one of the main tools in the theory of quasiconformal and quasiregularmappings.Theconformalmoduluscanbeusedtodefinequasiconfor- malmappingsintheplaneandinspace.Ithasalsobeenemployedinmetricmeasure O.Martioetal.,ModuliinModernMappingTheory,SpringerMonographsinMathematics, DOI10.1007/978-0-387-85588-2 1,(cid:2)c SpringerScience+BusinessMedia,LLC2009
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