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Analysis of quasiconformal maps in Rn

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Analysis of quasiconformal maps in Rn
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Purcell, Andrew
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Euclidean spaces
Bounded distortion
Moduli of curve families
Dilation
Absolute continuity on lines
Jacobian
Dissertations, Academic -- Mathematics -- Masters -- USF
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In this thesis, we examine quasiconformal mappings in Rn. We begin by proving basic properties of the modulus of curve families. We then give the geometric, analytic,and metric space definitions of quasiconformal maps and show their equivalence. We conclude with several computational examples.
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Thesis (M.A.)--University of South Florida, 2006.
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by Andrew Purcell.
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AnalysisofQuasiconformalMapsinRnbyAndrewPurcellAdissertationsubmittedinpartialfulllmentoftherequirementsforthedegreeofMasterofArtsDepartmentofMathematicsCollegeofArtsandSciencesUniversityofSouthFloridaMajorProfessor:ThomasBieske,Ph.D.ManougManougian,Ph.D.JogindarRatti,Ph.D.DateofApproval:April28,2006Keywords:Euclideanspaces,boundeddistortion,moduliofcurvefamilies,dilation,absolutecontinuityonlines,Jacobian.cCopyright2006,AndrewPurcell

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TableofContentsListofFiguresiiiAbstractiv1BackgroundandMotivation11.1IntroductionandMotivation.......................11.2Preliminaries...............................11.2.1MobiusSpace...........................11.2.2LinearAlgebra..........................21.2.3PartialDerivatives........................51.2.4Dierentiability..........................62TheModulusofaCurveFamily92.1TheGeometryofPaths..........................92.1.1Paths...............................92.1.2ChangeofParameter.......................102.1.3LineIntegrals...........................112.1.4ConformalMaps.........................122.2Thep-Modulus..............................132.2.1DenitionsandProperties....................132.2.2InuenceofNon-RectiableCurves...............172.2.3UpperandLowerBoundsforthep-Modulus..........192.2.4TheModulusofConformalMappings..............26i

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3QuasiconformalMappings293.1TheGeometricDenitionofQuasiconformalityandProperties....293.1.1TheDilatationofaHomeomorphism..............293.1.2TheDilatationofaLinearMap.................313.1.3QuasiconformalDieomorphisms................373.2MetricandAnalyticDenitionsandProperties............413.2.1TheLinearDilatation......................413.2.2TheACLProperty........................423.2.3TheMetricandAnalyticDenitionsofQuasiconformality..443.3EquivalenceoftheDenitions......................474ComputationalExplorations484.1RadialMappings.............................484.2Folding...................................504.3Cones...................................52References55ii

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ListofFigures1.1Thedilatationellipse...........................53.1Eccentricityofthelineardilatation...................323.2InscribedandcircumscribedballsofEA...............323.3Thecube..................................393.4Thelineardilatation...........................42iii

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AnalysisofQuasiconformalMapsinRnAndrewPurcellABSTRACTInthisthesis,weexaminequasiconformalmappingsinRn.Webeginbyprovingbasicpropertiesofthemodulusofcurvefamilies.Wethengivethegeometric,analytic,andmetricspacedenitionsofquasiconformalmapsandshowtheirequivalence.WeconcludewithseveralcomputationalExamples.iv

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1BackgroundandMotivation1.1IntroductionandMotivationWhenstudyingmappingsfromRnintoRn,itisdesirabletoconsidermapsthatdonotdistortthegeometryofthedomain.Themostnaturalchoiceofsuchmappingsisthosefunctionsthatmapcirclesintocircles.Suchmaps,calledconformalmaps,turnouttobetoorestrictiveintheirproperties.Atheoremby[L]showsthatsuchmappingsreducetorestrictionsoftranslations,dilationsorrotations.Becauseofthisrigidity,werelaxourgeometricconditionandconsiderthosemapswhoseimagesofcirclesareellipticalwithboundeddistortion.Inotherwords,theratiobetweenthemajorandminoraxisoftheimageiscontrolledbyaxedniteconstant.Thesemaps,calledquasiconformalmaps,provideaninterestingcourseofstudyandhavebeenconsideredinavarietyofmetricspaces.Inthisthesis,wewillfocusonquasiconformalmappingsinRnanddiscussvariousequivalentdenitions.Inaddition,weexaminevariouspropertiesandcomputationallyexploresomeExamplesinR3,enablingadeeperunderstandingofthemechanicsofquasiconformalmappings.Thematerialinthisthesisprovidesabasisforexplorationofquasiconformalmapsingeneralmetricspaces.1.2Preliminaries1.2.1MobiusSpaceLetRdenotetheeldofrealnumbers.Forn2,Rn=RR| {z }ntimesandx2Rnisgivenbyx=x1;x2;:::;xnwhereeachxi2R.Withtheusualadditionandscalar1

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multiplication,Rnbecomesan-dimensionalvectorspaceoverR.Thevectorse1=;0;0;:::;0;e2=;1;0;:::;0;...anden=;0;:::;0;1formabasisforRn.Nowforx;y2RnwedenetheEuclideaninnerproductofxandyashx;yi=x1y1+x2y2++xnynandtherelatedEuclideannormaskxk=q x21+x22++x2n:Inaddition,wehavethefollowingnotation:Bnx;r=fy2Rn:jy)]TJ/F22 11.955 Tf 11.955 0 Td[(xj
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arecalledthemaximalandminimalstretchingsofA.WedenotethecompositionoflineartransformsAandBbyAB.ForalineartransformA:Rn!RmandB:Rm!RpitisclearfromthedenitionsthatLABLALBand`AB`AlB:AlineartransformA:Rn!Rnissaidtobenon-singularifandonlyif`A>0,whichleadstotherelations:LA)]TJ/F20 7.97 Tf 6.586 0 Td[(1=`A)]TJ/F20 7.97 Tf 6.587 0 Td[(1and`A)]TJ/F20 7.97 Tf 6.586 0 Td[(1=LA)]TJ/F20 7.97 Tf 6.586 0 Td[(1:RecallfromModernAlgebrathenon-singularlineartransformsofRnformagroupwithcompositiondenotedbyGLn[R][Chapter2].AlineartransformA:Rn!RnissaidtobeanorthogonaltransformifjAxj=kxkforallx2RnorequivalentlyifhAx;Ayi=hx;yiforallx;y2Rn.Moreover,theorthogonaltransformsofRnformasubgroupofGLndenotedbyOn.ForeverylineartransformA,thereexistsauniquelineartransformA:Rn!RnsuchthathAx;yi=hx;Ayiforallx;y2Rn;wecallthelineartransformAtheadjointofA.NotethatifA2GLn,thenA2OnifandonlyifA)]TJ/F20 7.97 Tf 6.587 0 Td[(1=A:RecallthatifA2On,thenLA=1=`AandthatdetA=1.IfdetA=1,thenA2SOnthespecialorthogonalgroupwhichisasubgroupofOn.[R]NotingthatifA:Rn!Rmisalineartransform,andBandCareorthogonaltransforms,thenLCAB=LAand`CAB=`A:WhenalineartransformS:Rn!RnhasthepropertythatS=SwesaythatSissymmetricorself-adjoint.NotethatifA:Rn!RnisanarbitrarytransformbothAAandAAaresymmetric.Thefollowingtheoremconcerningsymmetricmatricescanbefoundin[HK].Theorem1.2.1IfSisasymmetriclineartransform,thenthereexistsanA2On3

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suchthatD=A)]TJ/F20 7.97 Tf 6.587 0 Td[(1SA,whereDx=1x1;2x2;:::;nxn.Moreover,theeigenvaluesofS,denoted1;:::;n,arereal.ThetransformDinTheorem1.2.1issaidtobeadiagonaltransformandfromLinearAlgebrawehavemin1injijjDxj=q 21x21++2nx2nwhenkxk=1.Hence,LD=maxjij;`D=minjij;anddetD=12n:ByanappropriatechoiceofAinTheorem1.2.1wealwaysassume12n.WesaythatalineartransformA:Rn!Rnispositivesemi-deniteorpositivedeniteifhx;Axi0orifhx;Axi>0forall06=x2Rn.Moreover,foranylineartransformA:Rn!Rn,AAandAAarealwayspositivesemi-denite.AsaconsequenceofTheorem1.2.1andthefactsabove,wehavethatifA:Rn!RnisalineartransformsuchthatAispositivesemi-deniteandsymmetric,theeigenvaluesofAcanbewrittenas12n0.WethenconcludeLA=LT)]TJ/F20 7.97 Tf 6.586 0 Td[(1AT=LD=1andsimilarly`A=n,whereT2OnsuchthatT)]TJ/F20 7.97 Tf 6.587 0 Td[(1AT=DasinTheorem1.2.1.Theorem1.2.2AnylineartransformA:Rn!RnmaybefactoredintoA=PB,whereB2OnandPisbothsymmetricandpositivesemi-denite.WhenconsideringthetransforminTheorem1.2.2weseePisuniquelydeterminedbyAandwehave:AA=PBPB=PBBP=PIP=P2;whereIisthennidentitymatrix.RecallBB=IsinceB2On.WecallPtheuniquesymmetric,positivesemi-denitesquarerootofAA.4

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Figure1.1:ThedilatationellipseThereforewemaysaythattheeigenvaluesofPare1 21;:::;1 2n,where12n0andeachiisaneigenvalueofAA.FromageometricperspectivethetransformAmapstheunitsphereSn)]TJ/F20 7.97 Tf 6.586 0 Td[(1toanellipsoidE,denotedbyEA[V].Example1.2.3LetT=24311335:ThetransformTmapsS1ontotheellipseETasshowninFigure1.1.ThetransformThaseigenvalues2and4,whichcorrespondtoeigenvectors0@)]TJ/F19 11.955 Tf 9.298 0 Td[(111Aand0@111A.WeconcludewithanextensionofTheorem1.2.1.Theorem1.2.4LetA:Rn!Rnbealineartransformandlet1;:::;nbetheeigenvaluesofAAsuchthat12n0,thenthereexistsU;V2OnsuchthatVAU=DwhereDx=1 21x1;:::;1 2nxn:1.2.3PartialDerivativesDeneafunctionf:ARn!Rm,andlet@ifdenotethepartialderivativeoffwithrespecttotheithcoordinate.Thatis,@ifx=limt!0fx+tei)]TJ/F22 11.955 Tf 11.955 0 Td[(fx t5

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forallx2A,whereAdenotestheinteriorofA,forwhichthelimitexistsinRm.Writingf=f1;f2;:::;fmwehave@ifxexistsifandonlyif@ifjxexistsforj=1;2;:::;m.Inparticular,@ifx=@if1x;:::;@ifmx.1.2.4DierentiabilityDenition1.2.5Letf:ARn!Rm.Wesayfisdierentiableatapointx,ifx2AandthereexistsalineartransformT:Rn!Rmsuchthatfy=fx+Ty)]TJ/F22 11.955 Tf 11.955 0 Td[(x+jy)]TJ/F22 11.955 Tf 11.955 0 Td[(xj"y1.2.1forally2Awithlimy!x"y=0:IfsuchatransformTexists,itisuniqueforeachfandforallh2RnTh=limh!0fx+th)]TJ/F22 11.955 Tf 11.955 0 Td[(fx t:.2.2TiscalledtheFrechetderivativeoffatxanddenotedbyf0xorDfx.Now,weletf=f1;:::;fmandconsidertheindividualcomponentsofthefunctioninEquation.2.1.Weseethatfisdierentiableatxifandonlyifeachfiisdierentiableatxandthatiffisdierentiablethenfiscontinuous.Iff0xexists,itimpliestheexistenceofthepartialderivativesoff.Moreover,Equation.2.2impliesthat@ifx=f0xei.Hence,thematrixf0xwithrespecttothestandardbasisforRnandRmis:@1f1x@2f1x@nf1x@1f2x@2f2x@nf2x.........@1fmx@2fmx@nfmx:6

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Theexistenceofeach@ifxdoesnotimplytheexistenceoff0x[S].However,ifeach@ifisdenedinsomeneighborhoodofxandeachiscontinuousatxthenfisdierentiableatx.Next,considerARnandletx2AwedenethemaximalstretchingLfxandtheminimalstretchinglfxatxofafunctionf:A!Rnby:Lfx=limsuph!0jfx+h)]TJ/F22 11.955 Tf 11.956 0 Td[(fxj jhj;lfx=liminfh!0jfx+h)]TJ/F22 11.955 Tf 11.956 0 Td[(fxj jhj:Theorem1.2.6Iffisdierentiableatx,thenLfx=Lf0xandlfx=lf0x:Proof.RecallthatifA:Rn!RmisalineartransformthenLA=maxkxk=1jAxjand`A=minkxk=1jAxj:"Now,hcloseto0impliesx+h2AandEquation.2.1yields:jfx+h)]TJ/F22 11.955 Tf 11.955 0 Td[(fxj=jf0xh+jhj"hjjhjLf0x+jhjj"hj:Hence,Lfxlimsuph!0Lf0x+j"hj=Lf0x:"Wemaypickh2Sn)]TJ/F20 7.97 Tf 6.587 0 Td[(1suchthatLf0x=jf0xhj.ThereforebyEquation.2.2,Lfxlimt!0jfx+th)]TJ/F22 11.955 Tf 11.955 0 Td[(fxj t=jf0xhj=Lf0x:Similarly,`fx=`f0x. Corollary1.2.7Iffisdierentiableatx,thenLfx=jf0xj.Theorem1.2.8LetARnandBRm.Iff:A!Rmisdierentiableatxandifg:B!Rpisdierentiableaty=fx,thenthecompositiongf:Af)]TJ/F20 7.97 Tf 6.587 0 Td[(1B!Rp7

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isdierentiableatx.Moreovergf0x=g0yf0x:8

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2TheModulusofaCurveFamilyInthischapterwewillexplorethenotionofthemodulusofacurvefamily.Thiswillbethemaintoolweemploywhendiscussingthepropertiesofquasiconformalmaps.2.1TheGeometryofPaths2.1.1PathsDenition2.1.1ApathincRnisacontinuousmap:!cRn,whereisanintervalinR.Thepathissaidtobeopenorcloseddependingonwhetherisopenorclosed.Thelocusjjofapath:!cRnisthesetfgcRn.Asubpathof:!cRnistherestrictionoftoacontinuoussubintervalof.Considerapartitionof[a;b]suchthata=t0t1:::tn=b.Wedenotethelengthof:[a;b]!cRnby`suchthat`=supfnXi=1jti)]TJ/F22 11.955 Tf 11.956 0 Td[(ti)]TJ/F20 7.97 Tf 6.586 0 Td[(1jg:.1.1Hence,0`1forallcRn.Clearly`=0ifandonlyifisaconstantpath.Denition2.1.2Wesaythepathisrectiableif`<1:Theorem2.1.3If:[a;b]!cRnisarectiablepath,andifa=t0t1:::tn=bisapartitionof[a;b],theneveryrestrictionj[ti)]TJ/F21 5.978 Tf 5.756 0 Td[(1;ti]isrectiable.9

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Moreover,`=nXi=1`j[ti)]TJ/F21 5.978 Tf 5.756 0 Td[(1;ti]:Let:[a;b]!cRnbearectiablepath.Forallt2[a;b]wesetst=`j[a;t],sometimesdenotedbyst.Wesaythefunctions:[a;b]!Risthelengthfunctionof.2.1.2ChangeofParameterDenition2.1.4Apath:[a;b]!cRnisobtainedfromapath:[c;d]!cRnbyanincreasing/decreasingchangeofparameterifthereexistsanincreasing/decreasingcontinuousmaph:[a;b]![c;d]suchthat=h:Inparticular,if
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Denition2.1.8Apath:!cRnislocallyrectiableifeverycompactsubpathofisrectiable.Wedene`=supf`g,overallcompactsubpathsof.Wenotethatonclosedpaths,thedenitionsoflengthareequivalent.Throughthenexttheorem,wemayextendcertainpathstoclosedpathswithoutchanginglength.Theorem2.1.9If:a;b!Rnisarectiableopenpath,thenthereexistsauniqueextensiontoaclosedpath:[a;b]!Rn.Moreover,`=`.Proof.Lett2a;b.Supposetothecontrary,limt!btdoesnotexist.Wethencanndapositivenumberrandasequenceofnumberst1r,forallj2N.Hence,`j[t1;uk]kXj=1juj)]TJ/F22 11.955 Tf 11.955 0 Td[(tjj>krforeveryk.Thisleadstoacontradiction,sinceisarectiablepath.Thelastassertionfollowsdirectlyfromtheexistenceoftheextension. Weconcludeourdiscussionoflengthwiththefollowingtheorem.Theorem2.1.10Let:a;b!Rnbeanopenpathsuchthatisabsolutelycontinuousoneveryclosedsubintervalofa;b,thenislocallyrectiableand`=Zbaj0tjdt:2.1.3LineIntegralsHenceforth,wewillassumeAcRnisaBorelsetand:A!R[f1gisanon-negativeBorelfunction.Let:[a;b]!Abeaclosedrectiablepath.Wedenethe11

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lineintegralofoverby:Zds=Z`00tdt;where0isreparameterizationoffromTheorem2.1.6.Wenotethattheintegralontheleft-handsideoftheequationexists,since0isanon-negativeBorelfunctionandtheintegralistheusualLebesgueintegral.Wehavethefollowingimportantresultconcerninglineintegralsoftheimageofapath.Theorem2.1.11LetUbeanopensetinRnandf:U!Rmbecontinuous.Also,let:!Ubealocallyrectiablepathsuchthatfisabsolutelyoneveryclosedsubpathof.Thenfislocallyrectiable.If:jfj!R[f1gisanon-negativeBorelfunction,thenZfdsZfxLfxjdxj:2.1.4ConformalMapsWhiletheheartofthisdiscussionliesinthetheoryofquasiconformalmaps,weneedtorstintroducethenotionofaconformalmap.Denition2.1.12LetD;D0bedomainsincRn.Ahomeomorphismf:D!D0isconformaliff2C1andifjf0xhj=jf0xjjhjforallx2Dandh2Rn:WenotethatifD;D0aredomainsincRn;ahomeomorphismf:D!D0isconfor-malwhenfjDnf1;f)]TJ/F21 5.978 Tf 5.756 0 Td[(11gisconformal.RecallalsothataC1homeomorphismfisconformalifandonlyifjf0xjn=jJx;fjforallx2D.AtheoremofLiouville[L]saysthatforn3everyconformalmapisaMobiustransformation,whereaMobiustransformationisamappingf:cRn!cRnsuchthatfisacompositionofanitenumberofthefollowingtransformations:1Translations:fx=x+a;foraxeda2R:2Stretchings:fx=rx;where0
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4InversioninasphereSa;r:fx=a+r2x)]TJ/F23 7.97 Tf 6.586 0 Td[(a jx)]TJ/F23 7.97 Tf 6.586 0 Td[(aj2:2.2Thep-ModulusInthissectionwewillpresentthemodulusofacurvefamilyincRn.Byacurvefamily,wemeantheelementsof)-326(arecurvesincRn.2.2.1DenitionsandPropertiesLetF\051bethesetofallnon-negativeBorelfunctions:Rn!R[f1gsuchthatZds1foralllocallyrectiablecurves2.Foreachp1wedenethep-modulusof)]TJ -360.953 -20.921 Td[(as:Mp\051=inf2F\051ZRnpdmandifF\051=;wesetMp\051=1.Fromthedenitionofthep-moduluswecanseethat0Mp\0511.Lemma2.2.1F)]TJ/F44 11.955 Tf 7.314 0 Td[(=;ifonlyif)]TJ/F44 11.955 Tf 11.499 0 Td[(containsaconstantpath.Proof.Dene:[a;b]!cRn)-382(tobeaconstantpath.ByachangeofvariabletechniqueZds=Zbaj0tjdt=0sothesetofadmissiblefunctionsof)-442(isempty.Ontheotherhand,if)-441(hasnonon-constantpathsthen1willalwaysbeinF\051.Inparticular,foralllocallyrectiable)]TJ/F28 11.955 Tf 78.283 -4.649 Td[(Zds=1`=1: 13

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Fortheremainderofthisdiscussionifp=nwewillwriteMp\051=M\051andsaythatM\051isthemodulusof)]TJ/F22 11.955 Tf 20.646 0 Td[(:Theorem2.2.2MpisanoutermeasureinthespaceofallcurvesincRn.Thatis,1Mp;=0:2)]TJ/F20 7.97 Tf 27.682 -1.794 Td[(1)]TJ/F20 7.97 Tf 7.314 -1.794 Td[(2Mp)]TJ/F20 7.97 Tf 11.867 -1.794 Td[(1Mp)]TJ/F20 7.97 Tf 11.867 -1.794 Td[(2:3Mp1[i=1)]TJ/F23 7.97 Tf 7.314 -1.794 Td[(i1Xi=1Mp)]TJ/F23 7.97 Tf 11.867 -1.794 Td[(i:Proof.1Notethat0belongstoF;.Therefore,Mp;=0.2If)]TJ/F20 7.97 Tf 33.323 -1.793 Td[(1)]TJ/F20 7.97 Tf 7.314 -1.793 Td[(2,thenF)]TJ/F20 7.97 Tf 53.323 -1.793 Td[(2F)]TJ/F20 7.97 Tf 19.506 -1.793 Td[(1.ThisimpliesMp)]TJ/F20 7.97 Tf 11.866 -1.793 Td[(1Mp)]TJ/F20 7.97 Tf 11.866 -1.793 Td[(2:3IfPiMp)]TJ/F23 7.97 Tf 11.866 -1.794 Td[(i=1,theresultisclear.Therefore,takePiMp)]TJ/F23 7.97 Tf 11.867 -1.794 Td[(i<1.Now,given">0,foreachipickisuchthatRRnpidmMp)]TJ/F23 7.97 Tf 11.866 -1.793 Td[(i+"2)]TJ/F23 7.97 Tf 6.586 0 Td[(i:Let=P1i=1pi1 p.Giventhecurve2Si)]TJ/F23 7.97 Tf 7.315 -1.793 Td[(i,wehave2)]TJ/F23 7.97 Tf 7.314 -1.793 Td[(i0forsome)]TJ/F23 7.97 Tf 54.841 -1.793 Td[(i0.Next,weseethatZds=Z1Xi=1pi1 pZi0ds1:Thus2FSi)]TJ/F23 7.97 Tf 7.314 -1.794 Td[(i,andwecomputeusingtheMonotoneConvergenceTheoremMp[i)]TJ/F23 7.97 Tf 7.314 -1.793 Td[(iZRnp=ZRn1Xi=1pi1Xi=1ZRnpi1Xi=1Mp)]TJ/F23 7.97 Tf 11.867 -1.793 Td[(i+1Xi=1"2)]TJ/F23 7.97 Tf 6.586 0 Td[(i=1Xi=1Mp)]TJ/F23 7.97 Tf 11.867 -1.793 Td[(i+": Denition2.2.3Let)]TJ/F20 7.97 Tf 7.314 -1.793 Td[(1and)]TJ/F20 7.97 Tf 7.315 -1.793 Td[(2becurvefamiliesincRnsuchthatforeach2)]TJ/F20 7.97 Tf 7.314 -1.793 Td[(2,hasasubcurvebelongingto)]TJ/F20 7.97 Tf 7.314 -1.793 Td[(1.Inthiscasewesaythat)]TJ/F20 7.97 Tf 7.314 -1.793 Td[(2minorizes)]TJ/F20 7.97 Tf 59.892 -1.793 Td[(1anddenoteit)]TJ/F20 7.97 Tf 7.315 -1.794 Td[(1)]TJ/F20 7.97 Tf 7.314 -1.794 Td[(2.Theorem2.2.4If)]TJ/F20 7.97 Tf 7.314 -1.793 Td[(1)]TJ/F20 7.97 Tf 7.314 -1.793 Td[(2,thenMp)]TJ/F20 7.97 Tf 11.867 -1.793 Td[(1Mp)]TJ/F20 7.97 Tf 11.867 -1.793 Td[(2.14

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Proof.IfMp)]TJ/F20 7.97 Tf 11.867 -1.793 Td[(1=1theresultisclear.Therefore,takeMp)]TJ/F20 7.97 Tf 11.866 -1.793 Td[(1<1,andpick2F)]TJ/F20 7.97 Tf 19.505 -1.793 Td[(1.If2)]TJ/F20 7.97 Tf 7.314 -1.793 Td[(2islocallyrectiableandif2)]TJ/F20 7.97 Tf 7.315 -1.793 Td[(1ispickedsuchthat,thenZdsZds1:Hence,2F)]TJ/F20 7.97 Tf 19.506 -1.793 Td[(2andsoMp)]TJ/F20 7.97 Tf 11.866 -1.793 Td[(2RRnpdm.ThereforeMp)]TJ/F20 7.97 Tf 11.867 -1.793 Td[(2Mp)]TJ/F20 7.97 Tf 11.867 -1.793 Td[(1bytakingtheinmumoverall2F)]TJ/F20 7.97 Tf 19.506 -1.793 Td[(1: Wenotethatthethep-modulusofacurvefamily)-321(islargeiftherearemanycurvesin)-327(orifthecurvesin)-326(areshort.Denition2.2.5Thecurvefamilies)]TJ/F20 7.97 Tf 7.314 -1.793 Td[(1;)]TJ/F20 7.97 Tf 7.314 -1.793 Td[(2;:::arecalledseparateifthereexistsdis-jointBorelsetsEi2Rnsuchthatif2)]TJ/F23 7.97 Tf 7.314 -1.793 Td[(iislocallyrectiable,thenZEcids=0:Lemma2.2.6If)]TJ/F20 7.97 Tf 7.314 -1.794 Td[(1;)]TJ/F20 7.97 Tf 7.315 -1.794 Td[(2;:::areseparate,thenMp1[i=1)]TJ/F23 7.97 Tf 7.315 -1.793 Td[(i=1Xi=1Mp)]TJ/F23 7.97 Tf 11.867 -1.793 Td[(i:Proof.ItsucestoshowthatPiMp)]TJ/F23 7.97 Tf 11.867 -1.793 Td[(iMp\051,where)-277(=Si)]TJ/F23 7.97 Tf 7.314 -1.793 Td[(i,sinceMpisanoutermeasure.IfMp\051=1,thentheresultisclear.Therefore,takeMp\051<1.Let2F\051anddenei=Ei,wherefEigisthecollectionofBorelsetsseparating)]TJ/F23 7.97 Tf 63.958 -1.794 Td[(i.Hence,forlocallyrectiable2)]TJ/F23 7.97 Tf 7.314 -1.794 Td[(iwehaveZEcids=0:Therefore,bytheMonotoneConvergenceTheorem:0ZEci=limk!1Zminf;kgEcilimk!1kZEcids=0:15

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Henceforall,ZEci=0:Wethenhave1Zds=ZEi+Ecids=Zids+ZEcids=Zids:Therefore,i2F)]TJ/F23 7.97 Tf 19.506 -1.794 Td[(i.Hence,usingdisjointnessXiMp)]TJ/F23 7.97 Tf 11.867 -1.793 Td[(iXiZRnpidm=XiZEipdm=ZSiEipdmZRnpdm:Takingtheinmumoverall2F\051impliesXiMp)]TJ/F23 7.97 Tf 11.866 -1.794 Td[(iMp\051: Theorem2.2.7If)]TJ/F20 7.97 Tf 7.314 -1.793 Td[(1;)]TJ/F20 7.97 Tf 7.314 -1.793 Td[(2;:::areseparateandif)]TJ/F25 11.955 Tf 10.635 0 Td[()]TJ/F23 7.97 Tf 7.314 -1.793 Td[(iforalli,thenMp\051XiMp)]TJ/F23 7.97 Tf 11.867 -1.793 Td[(i:Proof.ByLemma2.2.6andthefactthat)]TJ/F25 11.955 Tf 190.122 0 Td[(Si)]TJ/F23 7.97 Tf 7.314 -1.793 Td[(itheresultisclear. 16

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2.2.2InuenceofNon-RectiableCurvesFromthedenitionofthep-modulus,thecurveswhicharenotlocallyrectiablehavenoimpactonthep-modulus.Thatis,thep-modulusofcurvesthatarenotlocallyrectiableiszero.Hence,Mp\051=Mp)]TJ/F20 7.97 Tf 11.867 -1.794 Td[(0,where)]TJ/F20 7.97 Tf 56.288 -1.794 Td[(0isthefamilyofalllocallyrectiablecurvesin.Thisnexttheoremshowswhenpn,wemayrestrictourselvestorectiablecurves.Wedenethefollowingnotation:If)-475(isacurvefamilyincRn,wedenotetheFr\051tobethefamilyofallnon-negativeBorelfunctions:Rn!R[f1gsuchthatforallrectiable2)]TJ/F28 11.955 Tf -73.334 -21.088 Td[(Zds1:IngeneralF\051Fr\051.Theorem2.2.8If)]TJ/F44 11.955 Tf 11.498 0 Td[(isacurvefamilyincRn,thenforpn,Mp\051=inf2Fr)]TJ/F44 11.955 Tf 7.314 0 Td[(Zndm=Mp)]TJ/F23 7.97 Tf 11.866 -1.793 Td[(r:Proof.BythefactthatF\051Fr\051,wehaveMp\051Mp)]TJ/F23 7.97 Tf 11.867 -1.793 Td[(r:Nowtoshowthereverseinequality:Fixpn,anddenetheauxiliaryfunction0:cRn![0;1]by0x=8<:jxjn plnjxj)]TJ/F20 7.97 Tf 6.586 0 Td[(1jxj2;1jxj<2:17

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Werstcomputethespaceintegral.ZRnp0dm=Zjxj<21pdm+Zjxj21 jxjn plnjxj!pdm=2nn+ZZ121 jxjn plnjxj!prn)]TJ/F20 7.97 Tf 6.586 0 Td[(1drd n)]TJ/F19 11.955 Tf 11.955 0 Td[(1=2nn+!n)]TJ/F20 7.97 Tf 6.586 0 Td[(1Z12r)]TJ/F20 7.97 Tf 6.586 0 Td[(1lnr)]TJ/F23 7.97 Tf 6.586 0 Td[(pdr=2nn+!n)]TJ/F20 7.97 Tf 6.586 0 Td[(1lnr1)]TJ/F23 7.97 Tf 6.587 0 Td[(p 1)]TJ/F22 11.955 Tf 11.956 0 Td[(p12=2nn+!n)]TJ/F20 7.97 Tf 6.586 0 Td[(1ln21)]TJ/F23 7.97 Tf 6.587 0 Td[(p 1)]TJ/F22 11.955 Tf 11.955 0 Td[(p<1:RecallthatnisthevolumeoftheunitballinRnand!nisthesurfaceareaofthesphereinRn.WenowwishtoshowR0ds=1,foralllocallyrectiable2)-359(thatarenotrectiable.CaseI: isbounded.Leta=infx2jjf0xg>0:Hence,Z0dsZads=a`=1:CaseII: isunbounded.Pickx2jjsuchthatjxj2:WethenhaveZ0dsZ1jxj1 tn plntdtZ1jxj1 tlntdt=lnlntj1jxj=1:Nowlet2Fr\051.Forall">0,weset"=p+"pp01 p.Wewanttoshowthat"2F\051.18

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CaseA: isrectiable.Z"ds=Zp+"pp01 pds>Z1:CaseB: isnotrectiable.Z"dsZ"0ds=1:Hence,"2F\051,whichleadsusto:Mp\051ZRnp"dm=ZRnpdm+"pZRnp0dm:Letting"!0,weobtainMp\051ZRnpdm:Thetheoremfollowsbytakingtheinmumoverall2Fr\051. Corollary2.2.9If)]TJ/F23 7.97 Tf 7.314 -1.793 Td[(risthefamilyofallrectiablecurvesin)]TJ/F44 11.955 Tf 7.314 0 Td[(,thenMp)]TJ/F25 11.955 Tf 13.034 0 Td[(n)]TJ/F23 7.97 Tf 7.314 -1.793 Td[(r=0,whenpn:2.2.3UpperandLowerBoundsforthep-ModulusGivenacurvefamily)-327(incRn,becausewearetakingtheinmum,generallyitisonlypossibletondanupperbound.Precisely,picking2F\051producesMp\051ZpdmasdemonstratedbythenextExample.Theorem2.2.10Letp0.Let)]TJ/F44 11.955 Tf 12.042 0 Td[(beacurvefamilysuchthatif2)]TJ/F44 11.955 Tf 7.314 0 Td[(,theniscontainedinaBorelsetGcRnand`r>0forall2)]TJ/F44 11.955 Tf 7.314 0 Td[(,whereislocally19

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rectiable.Then,Mp\051mG rp:Proof.Dene:Rn!Rbyx=8<:1 rx2G;0x=2G:WecomputethatZds=1 r`1 rr=1:Hence2F\051,whichleadstoZRnp=ZG1 rp=mG rpMp\0510: InTheorem2.2.10weestablishedanupperboundforthep-modulusof,foraspeciccurvefamily.Thetaskofndingalowerboundisunfortunatelymuchmoredicult.Tondalowerbound,wemustchooseanarbitrary2F\051andforall2,ndaxedconstantKsothatRpK.ThisisusuallydonebyapplyingHolder'sinequalityandFubini'sTheorem.WeusethistechniqueinthenextthreeExamples.Example2.2.11TheCylinderLetEbeaBorelsetinRn)]TJ/F20 7.97 Tf 6.586 0 Td[(1andleth>0andp0.DeneG=fx2Rnjx1;:::;xn)]TJ/F20 7.97 Tf 6.587 0 Td[(12Eand0
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Foreachy2Elety:[0;h]!Gbeaverticalpathsuchthatyt=y+tenwhichimpliesy2.Let2F\051,implyingforalllocallyrectiable2)]TJ/F28 11.955 Tf -98.327 -21.087 Td[(Zds1:Ifp>1wehaveZdsp1andusingHolder'sinequalitywehave,1Zds=Z1dsZpds1 pZ1p p)]TJ/F21 5.978 Tf 5.756 0 Td[(1dsp)]TJ/F21 5.978 Tf 5.756 0 Td[(1 p:Therefore,for1pwehave1ZpdsZdsp)]TJ/F20 7.97 Tf 6.587 0 Td[(1=Zpdshp)]TJ/F20 7.97 Tf 6.587 0 Td[(1=Zh0y+tenpdthp)]TJ/F20 7.97 Tf 6.587 0 Td[(1;.2.2sinceforeachy2E,wehavey:[0;h]!G,whichsendseacht2[0;h]toy+ten.Now,integratebothsidesofEquation.2.2overE.Hence,weobtainZE1dmn)]TJ/F20 7.97 Tf 6.586 0 Td[(1ZEhp)]TJ/F20 7.97 Tf 6.587 0 Td[(1Zh0y+tenpdtdmn)]TJ/F20 7.97 Tf 6.587 0 Td[(1;whichisequivalenttomn)]TJ/F20 7.97 Tf 6.586 0 Td[(1Ehp)]TJ/F20 7.97 Tf 6.586 0 Td[(1ZEZh0y+tenpdtdmn)]TJ/F20 7.97 Tf 6.587 0 Td[(1;21

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whichisagainequivalenttomn)]TJ/F20 7.97 Tf 6.587 0 Td[(1E hp)]TJ/F20 7.97 Tf 6.586 0 Td[(1ZEZh0y+tenpdtdmn)]TJ/F20 7.97 Tf 6.587 0 Td[(1=ZGpdmnFubini'sTheoremZRnpdmn:Takingtheinmumoverall2F\051,weobtainmn)]TJ/F20 7.97 Tf 6.587 0 Td[(1E hp)]TJ/F20 7.97 Tf 6.586 0 Td[(1Mp\051:Hence,mn)]TJ/F20 7.97 Tf 6.587 0 Td[(1E hp)]TJ/F20 7.97 Tf 6.586 0 Td[(1Mp\051mnG hp=mn)]TJ/F20 7.97 Tf 6.586 0 Td[(1E hp)]TJ/F20 7.97 Tf 6.586 0 Td[(1:Example2.2.12TheSphericalRingLet0
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Lettingq=p)]TJ/F23 7.97 Tf 6.587 0 Td[(n p)]TJ/F20 7.97 Tf 6.587 0 Td[(1andc=Rbatq)]TJ/F20 7.97 Tf 6.586 0 Td[(1dt,wehave1Zbaptytn)]TJ/F20 7.97 Tf 6.586 0 Td[(1dtZbatq)]TJ/F20 7.97 Tf 6.586 0 Td[(1dtp)]TJ/F20 7.97 Tf 6.586 0 Td[(1=cp)]TJ/F20 7.97 Tf 6.586 0 Td[(1Zbaptytn)]TJ/F20 7.97 Tf 6.586 0 Td[(1dt:.2.3Therefore,1cp)]TJ/F20 7.97 Tf 6.587 0 Td[(1Zbaptytn)]TJ/F20 7.97 Tf 6.587 0 Td[(1dt:Note:c=8><>:Zbat1)]TJ/F24 5.978 Tf 5.757 0 Td[(n p)]TJ/F21 5.978 Tf 5.756 0 Td[(1dtp6=n;)]TJ/F19 11.955 Tf 5.479 -9.684 Td[(lnb ap=nandsowhenp6=nthenwehavec=Zbat1)]TJ/F24 5.978 Tf 5.756 0 Td[(n p)]TJ/F21 5.978 Tf 5.756 0 Td[(1dt=p)]TJ/F19 11.955 Tf 11.955 0 Td[(1 p)]TJ/F22 11.955 Tf 11.955 0 Td[(ntp)]TJ/F22 11.955 Tf 11.955 0 Td[(n p)]TJ/F19 11.955 Tf 11.955 0 Td[(1ba=p)]TJ/F19 11.955 Tf 11.955 0 Td[(1 p)]TJ/F22 11.955 Tf 11.955 0 Td[(nbp)]TJ/F24 5.978 Tf 5.756 0 Td[(n p)]TJ/F21 5.978 Tf 5.756 0 Td[(1)]TJ/F22 11.955 Tf 11.955 0 Td[(ap)]TJ/F24 5.978 Tf 5.756 0 Td[(n p)]TJ/F21 5.978 Tf 5.756 0 Td[(1:.2.4IntegratingEquation.2.3overy2Sn)]TJ/F20 7.97 Tf 6.587 0 Td[(1withrespecttosurfaceareayieldsZy2Sn)]TJ/F21 5.978 Tf 5.756 0 Td[(11dmn)]TJ/F20 7.97 Tf 6.587 0 Td[(1Zy2Sn)]TJ/F21 5.978 Tf 5.756 0 Td[(1cp)]TJ/F20 7.97 Tf 6.586 0 Td[(1Zbaptytn)]TJ/F20 7.97 Tf 6.586 0 Td[(1dtdmn)]TJ/F20 7.97 Tf 6.586 0 Td[(1:Weobtain!n)]TJ/F20 7.97 Tf 6.586 0 Td[(1cp)]TJ/F20 7.97 Tf 6.587 0 Td[(1ZSn)]TJ/F21 5.978 Tf 5.756 0 Td[(1Zbaptytn)]TJ/F20 7.97 Tf 6.587 0 Td[(1dtdmn)]TJ/F20 7.97 Tf 6.586 0 Td[(1=cp)]TJ/F20 7.97 Tf 6.587 0 Td[(1ZbaZSn)]TJ/F21 5.978 Tf 5.756 0 Td[(1ptytn)]TJ/F20 7.97 Tf 6.587 0 Td[(1dtdmn)]TJ/F20 7.97 Tf 6.586 0 Td[(1=cp)]TJ/F20 7.97 Tf 6.587 0 Td[(1ZApdmcp)]TJ/F20 7.97 Tf 6.586 0 Td[(1ZRnpdm:Takingtheinmumimplies!n)]TJ/F20 7.97 Tf 6.587 0 Td[(1cp)]TJ/F20 7.97 Tf 6.586 0 Td[(1Mp)]TJ/F23 7.97 Tf 11.866 -1.793 Td[(A:23

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Nowwewillshowthereverseinequality.Dene2F)]TJ/F23 7.97 Tf 19.506 -1.793 Td[(Abyx=8<:c)]TJ/F20 7.97 Tf 6.586 0 Td[(1jxjq)]TJ/F20 7.97 Tf 6.587 0 Td[(1x2A0x=2A:Nowforalllocallyrectiable2)]TJ/F23 7.97 Tf 7.315 -1.793 Td[(AZdsZbatdt=1 cZbatq)]TJ/F20 7.97 Tf 6.587 0 Td[(1dt=1:Therefore,2F)]TJ/F23 7.97 Tf 19.506 -1.793 Td[(A.Nowconsider,Mp)]TJ/F23 7.97 Tf 11.867 -1.793 Td[(AZRnpdm=ZbaZSn)]TJ/F21 5.978 Tf 5.757 0 Td[(1ptydmn)]TJ/F20 7.97 Tf 6.586 0 Td[(1tn)]TJ/F20 7.97 Tf 6.586 0 Td[(1dt=c)]TJ/F23 7.97 Tf 6.586 0 Td[(pZbaZSn)]TJ/F21 5.978 Tf 5.756 0 Td[(1tpq)]TJ/F20 7.97 Tf 6.586 0 Td[(1dmn)]TJ/F20 7.97 Tf 6.587 0 Td[(1tn)]TJ/F20 7.97 Tf 6.587 0 Td[(1dt=c)]TJ/F23 7.97 Tf 6.587 0 Td[(p!n)]TJ/F20 7.97 Tf 6.587 0 Td[(1Zbatpq)]TJ/F23 7.97 Tf 6.586 0 Td[(p+n)]TJ/F20 7.97 Tf 6.586 0 Td[(1dt=c)]TJ/F23 7.97 Tf 6.587 0 Td[(p!n)]TJ/F20 7.97 Tf 6.587 0 Td[(1Zbatq)]TJ/F20 7.97 Tf 6.586 0 Td[(1dt=c1)]TJ/F23 7.97 Tf 6.587 0 Td[(p!n)]TJ/F20 7.97 Tf 6.587 0 Td[(1:Wethenconclude!n)]TJ/F20 7.97 Tf 6.587 0 Td[(1c1)]TJ/F23 7.97 Tf 6.587 0 Td[(p=Mp)]TJ/F23 7.97 Tf 11.866 -1.793 Td[(A:24

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Example2.2.13TheDegenerateRingLet0
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Theorem2.2.15LetpnandletE;FandGbeBorelsets.MpfG:openjoinsEtoFg=MpfG:closedjoinsEtoFg:Proof.Let)-278(=fG:openjoinsEtoFgand)]TJ/F20 7.97 Tf 30.076 -1.793 Td[(0=fG:closedjoinsEtoFg.Since,)]TJ/F20 7.97 Tf 50.23 -1.793 Td[(0)-339(wehavebyTheorem2.2.4Mp)]TJ/F20 7.97 Tf 11.866 -1.793 Td[(0Mp\051.Toshowthereverseinequality,itsucestoshowF\051Fr)]TJ/F20 7.97 Tf 11.867 -1.793 Td[(0sinceMp\051=inf2Fr\051Zpdm:Let2F\051,letbearectiablepathin)]TJ/F20 7.97 Tf 129.619 -1.793 Td[(0,andletbetheclosedextensionofgivenbyTheorem2.1.9.Hence,jj=jjtouchesbothEandF.Therefore,thereexistst1andt2suchthatt12Eandt22F;wecantaket1t2.Nowdene=jt1;t22.Hence,Zds=ZdsZds1:Hence,2Fr)]TJ/F20 7.97 Tf 11.867 -1.793 Td[(0: 2.2.4TheModulusofConformalMappingsLetA2cRnandsupposef:A!cRnisacontinuousmap.If)-302(isafamilyofpathsinA,thenweset)]TJ/F26 7.97 Tf 76.64 4.339 Td[(0=ff:2)]TJ/F25 11.955 Tf 7.314 0 Td[(gandcall)]TJ/F26 7.97 Tf 51.536 4.339 Td[(0theimageof)-326(underf.Theorem2.2.16Iff:D!D0isconformal,thenM)]TJ/F26 7.97 Tf 11.867 4.338 Td[(0=M\051,forall)]TJ/F25 11.955 Tf 10.635 0 Td[(D.Proof.Let02F)]TJ/F26 7.97 Tf 19.506 4.338 Td[(0anddenex=0fxjf0xj.Wehave2F\051,sinceforalllocallyrectiable2)]TJ/F28 11.955 Tf -17.64 -21.087 Td[(Zds=Z0fxjf0xjds=Zf0xds1:26

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Nowsince2F\051,thisleadsusto:M\051ZDndm=ZD0fxjf0xjndm=ZD0fxnjf0xjndmsincefisconformal=ZD0fxnJx;fdm=ZD00fxndmZRn0ndm:Nowtakingtheinmum,weobtainM\051M)]TJ/F26 7.97 Tf 11.866 4.338 Td[(0.Werecallthatf)]TJ/F20 7.97 Tf 6.586 0 Td[(1isconformaliffisconformal.HenceM)]TJ/F26 7.97 Tf 11.866 4.339 Td[(0M\051andso,M)]TJ/F26 7.97 Tf 11.867 4.936 Td[(0=M\051: Itshouldbenotedthatthep-modulusisnotaconformalinvariantifp6=n.Inparticular,weintroduceaformulathatshowswhatwillhappentothep-modulusinaconformallinearmap.Theorem2.2.17Letc>0anddenef:Rn!Rnbyfx=cx.Denotetheimageofapathfamily)]TJ/F25 11.955 Tf 10.635 0 Td[(2Rnunderfbyc)]TJ/F44 11.955 Tf 7.315 0 Td[(.ThenMpc\051=cn)]TJ/F23 7.97 Tf 6.587 0 Td[(pMp\051Proof.fx=cximpliesthatjf0xj=candsincefisaconformallinearmapwenotethat`fx=Lfx=jf0xj=c:27

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Then,jJx;fj=cnforallx2Rn.ByTheorem2.2.16itsucestoshowcn)]TJ/F23 7.97 Tf 6.586 0 Td[(pMp\051Mpc\051orMp\051cp)]TJ/F23 7.97 Tf 6.587 0 Td[(nMpc\051:Let^2Fc\051anddenex=c^fx.Hence,1Zf^dsZ^fjf0xjds=Z^fcds=Zds:Thisimpliesthat2F\051,andsoMp\051ZRnpdm=ZRncp^fpdm=cp)]TJ/F23 7.97 Tf 6.587 0 Td[(nZRn^fpcndm=cp)]TJ/F23 7.97 Tf 6.586 0 Td[(nZRn^fpjJx;fjdm=cp)]TJ/F23 7.97 Tf 6.587 0 Td[(nZRn^pdm:Nowbytakingtheinmumoverall2F\051,weobtain:Mp\051cp)]TJ/F23 7.97 Tf 6.586 0 Td[(nMpc\051: 28

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3QuasiconformalMappingsInthischapterwewillgivethreedierentdenitionsofquasiconformalmapsdiscussedbyJ.Vaisala[V].Wewillthenexploredierentpropertiesofquasiconformalmaps.Lastly,wewilldemonstratetheequivalenceofthethreedenitions.3.1TheGeometricDenitionofQuasiconformalityandProperties3.1.1TheDilatationofaHomeomorphismLetf:D!D0beahomeomorphismandlet)-341(beacurvefamilyinD.Nowdenethequantities:KIf=supM)]TJ/F26 7.97 Tf 11.866 4.338 Td[(0 M\051andKOf=supM\051 M)]TJ/F26 7.97 Tf 11.866 3.454 Td[(0where)]TJ/F26 7.97 Tf 43.14 4.338 Td[(0=ff:2)]TJ/F25 11.955 Tf 7.314 0 Td[(gandsupremaaretakenoverall)-494(suchthatM\051andM)]TJ/F26 7.97 Tf 11.866 4.339 Td[(0arenotboth0or1.WesayKIfistheinnerdilatationoff,KOfistheouterdilatationoff,andKf=maxfKOf;KIfgisthemaximaldilatationoff.WenotethatKI1orKO1,henceK1.Denition3.1.1IfKf=K<1,wesayfisK-quasiconformalorK-qc.ThemapfisK-qcifandonlyifM\051 KM)]TJ/F26 7.97 Tf 11.866 4.936 Td[(0KM\051forall)]TJ/F25 11.955 Tf 10.635 0 Td[(D.WenotebyTheorem2.2.16,fisconformalifandonlyifM\051=M)]TJ/F26 7.97 Tf 11.866 4.339 Td[(0,whichimpliesthatK=1,inDenition3.1.1.29

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Theorem3.1.2Letf:D!D0beahomeomorphism.Thefollowingpropertiesholdforallx2D:1KIf=KOf)]TJ/F20 7.97 Tf 6.587 0 Td[(1;.1.12KOf=KIf)]TJ/F20 7.97 Tf 6.587 0 Td[(1;.1.23Kf)]TJ/F20 7.97 Tf 6.586 0 Td[(1=Kf;.1.34KIfgKIfKIg;.1.45KOfgKOfKOg;.1.56KfgKfKg:.1.6Proof.Let)-327(beafamilyofpathsinD.Bythedenitionof)]TJ/F26 7.97 Tf 119.671 4.338 Td[(0,theresultsareclearifanyofthedilatationsareinnite.Hence,wewillassumeKOandKItobenite.Recall:f:D!D0,)]TJ/F25 11.955 Tf 17.788 0 Td[(D,and)]TJ/F26 7.97 Tf 37.23 4.338 Td[(0=f)]TJ/F25 11.955 Tf 10.635 0 Td[(D0.ToshowRelation.1.1wenoteM)]TJ/F26 7.97 Tf 11.867 4.338 Td[(0 M\051=M)]TJ/F26 7.97 Tf 11.866 4.338 Td[(0 Mf)]TJ/F20 7.97 Tf 6.586 0 Td[(1)]TJ/F26 7.97 Tf 7.314 3.454 Td[(0KOf)]TJ/F20 7.97 Tf 6.586 0 Td[(1andbytakingthesupremumoverall)-352(weobtainKIfKOf)]TJ/F20 7.97 Tf 6.587 0 Td[(1.Interchangingtherolesof)-326(and)]TJ/F26 7.97 Tf 102.163 4.339 Td[(0weobtainRelation.1.2.Relation.1.3followsfromRelations3.1.1and.1.2.NowlookingatKIfgweseethat:M)]TJ/F26 7.97 Tf 11.866 4.339 Td[(0 M\051=Mfg\051 M\051=Mfg\051 Mg\051Mg\051 M\051KIfKIg:Takingthesupremumoverall)-327(givesusRelation.1.4.Relation.1.5followsinasimilarfashion.Relations.1.4and3.1.5togethergivesusKIfgKIfmaxfKOg;KIgg;30

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hence,KIfgKIfKg:Similarly,KOfgKOfKg.ThereforeKOfgKfKg;KIfgKfKg;andsoKfgKfKg. Corollary3.1.3IffisK-qc,thenf)]TJ/F20 7.97 Tf 6.587 0 Td[(1isK-qc.Corollary3.1.4Ifh=fg,wherefisK1-qcandgisK2-qc,thenhisK1K2-qc.3.1.2TheDilatationofaLinearMapLetA:Rn!Rnbealinearbijection,wedenethefollowingquantities:HIA=jdetAj `An;HOA=jAjn jdetAj;andHA=jAj `A:.1.7WesaythequantitiesHI;HO;andHaretheinner,outer,andlineardilatationsofA,respectively.InthenextsectionwewillshowthatHIA=KIAandHOA=KOA,whichexplainstheterminology.Ingeometricterms,HAmeasurestheoutofroundness"oreccentricityoftheellipsoidEA,seeFigure3.1[IM],whileHIAandHOArelatethevolumeofEBntothevolumesoftheinscribedandcircumscribedballscenteredaboutEA.ObservethatHIA=mEA mBIA;HOA=mBOA mEAandHAistheratioofthegreatestandsmallestsemi-axisofEA.SeeFigure3.2[IM].[Herea1a2:::anarethesemi-axisofEA.]31

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Figure3.1:Eccentricityofthelineardilatation Figure3.2:InscribedandcircumscribedballsofEA32

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WerecallthatthenumbersaiarethepositivesquarerootsoftheeigenvaluesofAA,whereAistheadjointofA.Wealsorecalla1=jAj;an=`A;anddetA=a1a2an:BythedenitionofthedilatationswecanwriteHOA=an)]TJ/F20 7.97 Tf 6.586 0 Td[(11 a2an;HIA=a1an)]TJ/F20 7.97 Tf 6.587 0 Td[(1 an)]TJ/F20 7.97 Tf 6.587 0 Td[(1n;andHA=a1 an:.1.8Theorem3.1.5IfA:Rn!Rnisalinearbijection,then1HIAHOAn)]TJ/F20 7.97 Tf 6.586 0 Td[(1;2HOAHIAn)]TJ/F20 7.97 Tf 6.586 0 Td[(1;3HAn=HIAHOA;4HAminfHIA;HOAgHAn 2maxfHIA;HOAgHAn)]TJ/F20 7.97 Tf 6.587 0 Td[(1:Proof.Leta1a2anbethesemi-axisofEA.1Wehavea1an)]TJ/F20 7.97 Tf 6.587 0 Td[(1an)]TJ/F20 7.97 Tf 6.587 0 Td[(11.Fromthisweobtain:a1an)]TJ/F20 7.97 Tf 6.587 0 Td[(1an)]TJ/F20 7.97 Tf 6.587 0 Td[(11a2an)]TJ/F20 7.97 Tf 6.587 0 Td[(1an)]TJ/F20 7.97 Tf 6.587 0 Td[(21a2an)]TJ/F20 7.97 Tf 6.586 0 Td[(1nan)]TJ/F20 7.97 Tf 6.586 0 Td[(21n=an2)]TJ/F20 7.97 Tf 6.587 0 Td[(2n1a1a2an)]TJ/F20 7.97 Tf 6.586 0 Td[(1nan)]TJ/F20 7.97 Tf 6.587 0 Td[(1na1an2)]TJ/F20 7.97 Tf 6.586 0 Td[(2n1an)]TJ/F20 7.97 Tf 6.587 0 Td[(1na1a2an)]TJ/F20 7.97 Tf 6.586 0 Td[(1nan)]TJ/F20 7.97 Tf 6.586 0 Td[(1nan2)]TJ/F20 7.97 Tf 6.587 0 Td[(2n+11an)]TJ/F20 7.97 Tf 6.587 0 Td[(1n=an)]TJ/F20 7.97 Tf 6.586 0 Td[(11n)]TJ/F20 7.97 Tf 6.587 0 Td[(1an)]TJ/F20 7.97 Tf 6.587 0 Td[(1na1an)]TJ/F20 7.97 Tf 6.587 0 Td[(1a2ann)]TJ/F20 7.97 Tf 6.586 0 Td[(1an)]TJ/F20 7.97 Tf 6.586 0 Td[(11n)]TJ/F20 7.97 Tf 6.586 0 Td[(1an)]TJ/F20 7.97 Tf 6.587 0 Td[(1na1an)]TJ/F20 7.97 Tf 6.587 0 Td[(1 an)]TJ/F20 7.97 Tf 6.587 0 Td[(1nan)]TJ/F20 7.97 Tf 6.586 0 Td[(11 a2ann)]TJ/F20 7.97 Tf 6.587 0 Td[(1HIAHOAn)]TJ/F20 7.97 Tf 6.587 0 Td[(1:33

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2Wehavean)]TJ/F20 7.97 Tf 6.587 0 Td[(1na2an.Fromthisweobtain:an)]TJ/F20 7.97 Tf 6.586 0 Td[(2na2an)]TJ/F20 7.97 Tf 6.587 0 Td[(1an)]TJ/F20 7.97 Tf 6.586 0 Td[(2nna2an)]TJ/F20 7.97 Tf 6.587 0 Td[(1nan)]TJ/F20 7.97 Tf 6.587 0 Td[(11an2)]TJ/F20 7.97 Tf 6.586 0 Td[(2nnanan)]TJ/F20 7.97 Tf 6.587 0 Td[(11a2an)]TJ/F20 7.97 Tf 6.587 0 Td[(1nanan)]TJ/F20 7.97 Tf 6.587 0 Td[(11an2)]TJ/F20 7.97 Tf 6.586 0 Td[(2n+1na2ana1an)]TJ/F20 7.97 Tf 6.587 0 Td[(1n)]TJ/F20 7.97 Tf 6.586 0 Td[(1an)]TJ/F20 7.97 Tf 6.586 0 Td[(11an)]TJ/F20 7.97 Tf 6.587 0 Td[(1nn)]TJ/F20 7.97 Tf 6.587 0 Td[(1a2ana1an)]TJ/F20 7.97 Tf 6.587 0 Td[(1n)]TJ/F20 7.97 Tf 6.586 0 Td[(1an)]TJ/F20 7.97 Tf 6.587 0 Td[(11 a2ana1an)]TJ/F20 7.97 Tf 6.587 0 Td[(1n)]TJ/F20 7.97 Tf 6.587 0 Td[(1 an)]TJ/F20 7.97 Tf 6.587 0 Td[(1nn)]TJ/F20 7.97 Tf 6.586 0 Td[(1an)]TJ/F20 7.97 Tf 6.586 0 Td[(11 a2ana1an)]TJ/F20 7.97 Tf 6.587 0 Td[(1 an)]TJ/F20 7.97 Tf 6.587 0 Td[(1nn)]TJ/F20 7.97 Tf 6.587 0 Td[(1HOAHIAn)]TJ/F20 7.97 Tf 6.587 0 Td[(1:3HAn=a1 ann=an1 ann=an1 anna2an)]TJ/F20 7.97 Tf 6.587 0 Td[(1 a2an)]TJ/F20 7.97 Tf 6.587 0 Td[(1=a1an)]TJ/F20 7.97 Tf 6.586 0 Td[(1 an)]TJ/F20 7.97 Tf 6.586 0 Td[(1nan)]TJ/F20 7.97 Tf 6.586 0 Td[(11 a2an=HIAHOA:4HAminfHIA;HOAg:Wewillproceedinparts:CaseI: HAHIA.Wehavean)]TJ/F20 7.97 Tf 6.586 0 Td[(1na2ana1an)]TJ/F20 7.97 Tf 6.586 0 Td[(1na1an=a1an)]TJ/F20 7.97 Tf 6.587 0 Td[(1ana1 ana1an)]TJ/F20 7.97 Tf 6.587 0 Td[(1 an)]TJ/F20 7.97 Tf 6.587 0 Td[(1nHAHIA:34

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CaseII: HAHOA.Wehavea1an)]TJ/F20 7.97 Tf 6.586 0 Td[(1an)]TJ/F20 7.97 Tf 6.587 0 Td[(11a1an)]TJ/F20 7.97 Tf 6.587 0 Td[(1anan)]TJ/F20 7.97 Tf 6.587 0 Td[(11ana1 anan)]TJ/F20 7.97 Tf 6.586 0 Td[(11 a2anHAHOA:WethenconcludeHAminfHIA;HOAg.WenextshowmaxfHIA;HOAgHAn)]TJ/F20 7.97 Tf 6.586 0 Td[(1:CaseI: HIAHAn)]TJ/F20 7.97 Tf 6.586 0 Td[(1.a1an)]TJ/F20 7.97 Tf 6.587 0 Td[(1an)]TJ/F20 7.97 Tf 6.587 0 Td[(11a1an)]TJ/F20 7.97 Tf 6.586 0 Td[(1 an)]TJ/F20 7.97 Tf 6.586 0 Td[(1nan)]TJ/F20 7.97 Tf 6.586 0 Td[(11 an)]TJ/F20 7.97 Tf 6.586 0 Td[(1n=a1 ann)]TJ/F20 7.97 Tf 6.587 0 Td[(1HIAHAn)]TJ/F20 7.97 Tf 6.586 0 Td[(1:CaseII: HOAHAn)]TJ/F20 7.97 Tf 6.587 0 Td[(1.an)]TJ/F20 7.97 Tf 6.586 0 Td[(1na2anan)]TJ/F20 7.97 Tf 6.586 0 Td[(11an)]TJ/F20 7.97 Tf 6.586 0 Td[(1nan)]TJ/F20 7.97 Tf 6.586 0 Td[(11a2anan)]TJ/F20 7.97 Tf 6.586 0 Td[(11 a2anan)]TJ/F20 7.97 Tf 6.587 0 Td[(11 an)]TJ/F20 7.97 Tf 6.587 0 Td[(1n=a1 ann)]TJ/F20 7.97 Tf 6.586 0 Td[(1HOAHAn)]TJ/F20 7.97 Tf 6.587 0 Td[(1:WethenconcludemaxfHIA;HOAgHAn)]TJ/F20 7.97 Tf 6.587 0 Td[(1:ItremainstoshowminfHIA;HOAgHAn 2maxfHIA;HOAg:35

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Thisfollowseasilyfrompart3,whichimpliesminfHIA;HOAg2HAnmaxfHIA;HOAg2: Example3.1.6IfthelineartransformAisgivenbyamatrix,itisusuallyadiculttasktocomputethedilatationsofA.Howeverwhenn=2,thisisaccomplishedbyaeasilyobtainedformula.LetA:R2!R2bealinearbijection,nowusingcomplexnotationinR2denethemapAz=ax+by+icx+dy,wherea;b;c;darerealnumberssuchthatad)]TJ/F22 11.955 Tf 12.107 0 Td[(bc6=0.Sincen=2,HA=HIA=HOA.LetH=HA,wenowcomputeHintermsofa;b;c;d.LetA=abcdbethematrixrepresentationofA.ThenAT=acbd,andATA=acbdabcd=a2+c2ab+cdab+cdb2+d2:NowdetI)]TJ/F22 11.955 Tf 11.955 0 Td[(ATA=det)]TJ/F19 11.955 Tf 11.955 0 Td[(a2+c2)]TJ/F22 11.955 Tf 9.299 0 Td[(ab)]TJ/F22 11.955 Tf 11.955 0 Td[(cd)]TJ/F22 11.955 Tf 9.298 0 Td[(ab)]TJ/F22 11.955 Tf 11.955 0 Td[(cd)]TJ/F19 11.955 Tf 11.955 0 Td[(b2+d2=)]TJ/F19 11.955 Tf 11.956 0 Td[(a2+c2)]TJ/F19 11.955 Tf 11.955 0 Td[(b2+d2)]TJ/F19 11.955 Tf 11.955 0 Td[()]TJ/F22 11.955 Tf 9.299 0 Td[(ab)]TJ/F22 11.955 Tf 11.956 0 Td[(cd)]TJ/F22 11.955 Tf 9.298 0 Td[(ab)]TJ/F22 11.955 Tf 11.955 0 Td[(cd=2)]TJ/F22 11.955 Tf 11.956 0 Td[(a2+b2+c2+d2+ad)]TJ/F22 11.955 Tf 11.955 0 Td[(bc2:36

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Nowlet1and2besolutionstodetI)]TJ/F22 11.955 Tf 11.955 0 Td[(ATA=0.Hence,1+2=a2+b2+c2+d2;12=ad)]TJ/F22 11.955 Tf 11.955 0 Td[(bc2;H=r 1 2;andH+1 H=r 1 2+r 2 1=1+2 p 12=a2+b2+c2+d2 jad)]TJ/F22 11.955 Tf 11.955 0 Td[(bcj:ThereforeHisthegreaterrootofthisequation,theotherrootisH)]TJ/F20 7.97 Tf 6.586 0 Td[(1.3.1.3QuasiconformalDieomorphismsInthissectionwewillshowthatthedilatationsofadieomorphismcanbederivedintermsofitsderivative.Consideradieomorphismf:D!D0,whereDandD0aredomainsinRn.WerecallthatadieomorphismisaC1homeomorphism,whoseJacobiandoesnotvanish.Thatis,Jx;f6=0forallx2D:Wenotethatiffisadieomorphism,byEquation.1.7wehave:HOf0x=jf0xjn jJx;fjandHIf0x=jJx;fj `f0xn:Theorem3.1.7Letf:D!D0beahomeomorphism.Iffisdierentiableatthepointa2DandifKOf<1,thenjf0ajnKOfjJa;fj:Proof.Withoutlossofgeneralitywemaytakea=0=fa,sinceifnot,wemayconsiderthemapg:fx2Rn:x+a2Dg!D0denedbygx=fx+a)]TJ/F22 11.955 Tf 11.516 0 Td[(fa.Therefore,g0=f0a.SobyTheorem3.1.2,wehaveKOfa=KOg.We37

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mayalsotakef0tobeoftheformf0x=a1x1;:::;anxnT;wherea1a2:::an.Hence,jf0j=a1;andjJ;fj=a1an:Wethenneedtoverifyan1KOfa1an:Theresultisclearwhena1=0,hencewetakea1>0.Now,forallx2Dwecanwritefx=f0x+jxj"xwherelimx!0"x=0.Now,let"2;a1 2andchoose>0suchthatthen-intervalQ=[0;][0;][0;]iscontainedDandsuchthatj"xjn)]TJ/F21 5.978 Tf 7.782 3.258 Td[(1 2"holdsforallx2Q.So,jfx)]TJ/F22 11.955 Tf 11.955 0 Td[(f0xj=jxjj"xj<":.1.9Now,letEandFbethefacesofQonwhichx0=0andx1=.Denethecurvefamily)-278(=fQ:joinsEandFg:ByExample2.2.11,M\051=1.WenextestimateM)]TJ/F26 7.97 Tf 11.867 4.338 Td[(0.ByEquation.1.9wehavefQG=fx:)]TJ/F22 11.955 Tf 9.298 0 Td[("xiai+"gandbyTheorem2.2.10M)]TJ/F26 7.97 Tf 11.867 4.936 Td[(0a1+2"an+2"a1)]TJ/F19 11.955 Tf 11.955 0 Td[(2")]TJ/F23 7.97 Tf 6.586 0 Td[(n:38

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Figure3.3:ThecubeCombiningtheestimatesofM\051andM)]TJ/F26 7.97 Tf 11.866 4.339 Td[(0,wehavea1)]TJ/F19 11.955 Tf 11.955 0 Td[(2"nM\051 M)]TJ/F26 7.97 Tf 11.866 3.454 Td[(0a1+2"an+2":Letting"!0yieldsan1M\051 M)]TJ/F26 7.97 Tf 11.867 3.454 Td[(0a1an;oran1KOfa1an: AnapplicationofTheorem3.1.7isthefollowing:Theorem3.1.8Supposef0a=0wheneverJa;f=0.Iff:D!D0isadieomorphism,thenKIf=supx2DHIf0x;andKOf=supx2DHOf0x:Proof.Therstequationfollowsfromthesecondbyapplyingtheinversemappingf)]TJ/F20 7.97 Tf 6.587 0 Td[(1,henceitsucestoonlyshowthesecondequation.39

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LetsupHOf0x=K<1.WemustshowKOf=M\051 M)]TJ/F27 5.978 Tf 8.581 2.27 Td[(0Kforanarbitrarypathfamily)-326(inD.Let02F)]TJ/F26 7.97 Tf 19.506 4.338 Td[(0.Now,dene:Rn!R[f1gbyx=8<:0fxjf0xjx2D0x=2D:Supposethatisalocallyrectiablepathin.By[V][Corollary5.4]weobtain:ZdsZf0ds1:Therefore,2F\051.Thisleadsusto:M\051ZDndm=ZD0fxnjf0xjndmKZD0fxnjJx;fjdm=KZfD0ndmKZRn0ndm:TakingtheinmumweobtainM\051KM)]TJ/F26 7.97 Tf 11.866 4.338 Td[(0:Thereverseinequalityisaconse-quenceofTheorem3.1.7. Corollary3.1.9Thefunctionf:D!D0isaK-qcdieomorphismifandonlyifjf0xjn KjJx;fjK`f0xnforallx2D.Moreover,1KIfKOfn)]TJ/F20 7.97 Tf 6.586 0 Td[(1and1KOfKIfn)]TJ/F20 7.97 Tf 6.587 0 Td[(1:AsadirectconsequenceofTheorem3.1.5andTheorem3.1.8wehavethefollowingcorollary.Corollary3.1.10Ifaqc-mappingfisdierentiableatapointa,theneitherf0a=0orJa;f6=0.40

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3.2MetricandAnalyticDenitionsandProperties3.2.1TheLinearDilatationInthissectionwewilldenethelineardilatationofahomeomorphism.LetDandD0bedomainsincRn,anddeneahomeomorphismf:D!D0.Foreachx2Dwithfx6=1,andforeachr>0suchthatSn)]TJ/F20 7.97 Tf 6.587 0 Td[(1x;rDwedenethelineardilatationsasfollows:Denition3.2.1SetthequantitiesLx;f;r=maxjy)]TJ/F23 7.97 Tf 6.586 0 Td[(xj=rjfy)]TJ/F22 11.955 Tf 11.955 0 Td[(fxj;andlx;f;r=minjy)]TJ/F23 7.97 Tf 6.586 0 Td[(xj=rjfy)]TJ/F22 11.955 Tf 11.955 0 Td[(fxj:Thelineardilatationoffatapointx2DisHx;f=limsupr!0Lx;f;r lx;f;r:Ifx=1,suchthatfx6=1,wesetHx;f=H;fu,whereuistheinversionmapux=x kxk2.Also,iffx=1wedeneHx;f=x;uf.Sincelx;f;rLx;f;r,wehave1Hx;f1:NowifAisabijectivelinearmap,wenotethatHx;A=HAforallx2Rn,whereHAisdenedasinEquation.1.7.WealsonotebyTheorem1.2.6,thatiffisdierentiableatxandifjJx;fj6=0,thenHx;f=Hf0x.Weomittheverytechnicalproofofthefollowingtheorem.Fordetailssee[V][pp.78-80].Theorem3.2.2Letf:D!D0beahomeomorphismsuchthatforsomeK<1eitherKOfKorKIfK.ThenHx;fisboundedbyaconstantwhichdependsonlyonnandK.41

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Figure3.4:ThelineardilatationCorollary3.2.3Iff:D!D0isqc,thenHx;fisbounded.3.2.2TheACLPropertyThissubsectionismodeledafterSections22and31of[V].WesetRn)]TJ/F20 7.97 Tf 6.586 0 Td[(1i=fx2Rn:xi=0g.InparticularPiistheprojectionofRnontoRn)]TJ/F20 7.97 Tf 6.586 0 Td[(1i.Thatis,Pix=x)]TJ/F22 11.955 Tf 11.955 0 Td[(xiei.Denition3.2.4LetQ=fx2Rn:x2[ai;bi]gbeaclosedn-interval.Amappingf:Q!RmissaidtobeabsolutelycontinuousonlinesACLiffiscontinuousandfisabsolutelycontinuousonalmosteverylinesegmentinQparalleltothecoordinateaxes.WenotethatifEiisthesetofallx2PiQsuchthatx7!fx+teiisnotabsolutelycontinuouson[ai;bi],thenmn)]TJ/F20 7.97 Tf 6.587 0 Td[(1Ei=0for1in.WealsosayifUisanopensetinRn,thenthemappingf:U!RmisACLifthemapfrestrictedtoQisACLforallclosedintervalsQU.Denition3.2.5AnACLmappingf:U!RmissaidtobeACLp,forp1,if@f @xi2LplocUforall1in.ItiswellknownthatACLp=W1;p,see[Z][2.1.4].Theorem3.2.6[V][Theorem31.2]Letf:D!D0beahomeomorphismsuchthatHx;fisbounded,thenfisACL.42

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Corollary3.2.7EveryqcmapisACL.Theorem3.2.8[V][Theorem32.1]Letf:D!D0beahomeomorphismsuchthatHx;fisbounded.Thenfisdierentiablealmosteverywhere.Corollary3.2.9Aqcmapisdierentiablealmosteverywhere.Theorem3.2.10[V][Theorem32.3]Letf:DcRn!D0beahomeomorphism.If1K1,thenthefollowingareequivalent:1KOfK:2fisACL,almosteverywheredierentiable,andjf0xjnKjJx;fja:e:Moreover,fisACLnwheneverandhold.Proof.AssumeKOfK.ByTheorem3.2.2,Hx;fisbounded.ByTheorems3.2.6and3.2.8,fisACLanda.e.dierentiable.Theinequalityin2followsfromTheorem3.1.7.Next,chooseEtobeacompactsetinDnf1;f)]TJ/F20 7.97 Tf 6.587 0 Td[(11g.WeseethatZEjf0xjndmKZEjJx;fjdmKmfE<1:Notingthatj@ifxjjf0xjateverypointxofdierentiability,wesee@if2Ln,andthuswehavefisACLn.Nowassumetheconditionsin2aresatised.Let)]TJ/F20 7.97 Tf 271.56 -1.793 Td[(0bethefamilyofalllocallyrectiablepaths2)]TJ/F25 11.955 Tf 12.776 0 Td[(Dsuchthatfisabsolutelycontinuousoneveryclosedsubpathofandlet)]TJ/F26 7.97 Tf 50.189 -1.793 Td[(1=f2)-966(:12fg.HencebyExample2.2.13M)]TJ/F26 7.97 Tf 11.866 -1.793 Td[(1=0.SincefisACLn,Fuglede'sTheorem[F]impliesM)]TJ/F25 11.955 Tf 14.623 0 Td[(n)]TJ/F26 7.97 Tf 11.867 -1.793 Td[(1n)]TJ/F20 7.97 Tf 7.315 -1.793 Td[(0=0.ThereforeM)]TJ/F20 7.97 Tf 11.866 -1.793 Td[(0=M\051,anditsucestoshowthatM)]TJ/F20 7.97 Tf 11.866 -1.793 Td[(0KM)]TJ/F26 7.97 Tf 11.866 4.338 Td[(0:RecallLfx=limsuph!0jfx+h)]TJ/F22 11.955 Tf 11.955 0 Td[(fxj jhj:43

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Let02F)]TJ/F26 7.97 Tf 19.506 4.339 Td[(0,andforx2Ddene:Rn!R[f1gbyx=8<:0xLfxx2D0x=2D:If2)]TJ/F20 7.97 Tf 7.314 -1.793 Td[(0,Theorem2.1.11givesusZdsZf0ds1:Hence2F)]TJ/F20 7.97 Tf 19.506 -1.793 Td[(0,whichimpliesM)]TJ/F20 7.97 Tf 11.866 -1.794 Td[(0ZDndm=ZD0fxnLfxndm=ZD0fxnjf0xjndmKZD0fxnjJx;fjdmKZRn0ndm:Sincethisholdsforall02F)]TJ/F26 7.97 Tf 19.506 4.338 Td[(0,thisimpliesthatM)]TJ/F20 7.97 Tf 11.867 -1.794 Td[(0KM)]TJ/F26 7.97 Tf 11.866 4.338 Td[(0: Corollary3.2.11AqcmapisACLn.3.2.3TheMetricandAnalyticDenitionsofQuasiconformalityLemma3.2.12Iff:D!D0isahomeomorphism,suchthatHx;fisboundedbyaconstantC,thenjf0xjnCn)]TJ/F20 7.97 Tf 6.587 0 Td[(1jJx;fj:Proof.ByTheorem3.2.6,fisACL,andbyTheorem3.2.8fisdierentiablealmosteverywhere.UsingCorollary3.1.10andTheorem1.2.6wehavethatforx2Deitherf0x=0or0
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Hence,jf0xjn jJx;fj=HOf0xmaxfHIf0;HOf0gHf0n)]TJ/F20 7.97 Tf 6.587 0 Td[(1=Cn)]TJ/F20 7.97 Tf 6.586 0 Td[(1;orjf0xjnCn)]TJ/F20 7.97 Tf 6.587 0 Td[(1jJx;fj: Theorem3.2.13[TheMetricDenitionofQuasiconformality]Ahomeomorphismf:D!D0isqcifandonlyifHx;fisbounded.Proof.Supposefisqc.ThenHx;fisboundedbyCorollary3.2.3.Nowsupposeforallx2D;Hx;f=C<1.Now,byTheorem3.2.6fisACL,andbyTheorem3.2.8fisdierentiablealmosteverywhere.So,forx2D,f0x=0or0
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D0beahomeomorphism.ThenfisK-qcifandonlyifthefollowingaresatised:1fisACL,2fisdierentiablealmosteverywhere,3Foralmosteveryx2Djf0xjn KjJx;fjK`f0xn:Proof.SupposefisK-qc.ByCorollary3.2.7fisACL,andbyCorollary3.2.9fisdierentiablealmosteverywhere.Henceconditions1and2holdtrue.Now,fisK-qcimpliesthatjf0xjn jJx;fjKforallx2Dsuchthatfisdierentiableatx.WemayalsotakejJx;fj6=0.Theinverseg=f)]TJ/F20 7.97 Tf 6.586 0 Td[(1isalsoK)]TJ/F22 11.955 Tf 11.955 0 Td[(qcandhasaderivativeaty=fx.Hence,jJx;fj=jJy;gj)]TJ/F20 7.97 Tf 6.586 0 Td[(1Kjg0yjn=K`f0xn:Nowassumethethreeconditionstoholdtrue.Thatis,fisanACLmap,dierentiablealmosteverywhere,andatalmostallx2Dwehave:jf0xjn KjJx;fjK`f0xn:WetakenotethatKOf
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3.3EquivalenceoftheDenitionsTheorem3.3.1Letf:DcRn!D0beahomeomorphism.Forall)]TJ/F25 11.955 Tf 12.341 0 Td[(Dthefollowingareequivalent:11 KM\051M)]TJ/F26 7.97 Tf 11.867 4.936 Td[(0KM\051:2fisACL,almosteverywheredierentiable,andjf0xjn KjJx;fjK`f0xn:3Hx;fisbounded.Proof.ByTheorems3.2.13and3.2.15theresultisclear. 47

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4ComputationalExplorationsInthissectionwegivesomeExamplesofquasiconformalmaps.4.1RadialMappingsExample4.1.1Let06=a2R,andletf:Rnnf0g!Rnnf0gbeadieomorphismdenedbyfx=jxja)]TJ/F20 7.97 Tf 6.587 0 Td[(1x:Wecanextendftoahomeomorphismf:cRn!cRnbydeningf=0,f1=1ifa>0,andf=1,f1=0ifa<0.Noteifa=1,fistheidentitymap.Ifa=)]TJ/F19 11.955 Tf 9.299 0 Td[(1,fisconformalandfistheinversionintheunitsphereSn)]TJ/F20 7.97 Tf 6.587 0 Td[(1.WerecallthatHOf0x=jf0xjn jJx;fj=an)]TJ/F20 7.97 Tf 6.586 0 Td[(11 a2a3anandHIf0x=jJx;fj `f0xn=a1a2an)]TJ/F20 7.97 Tf 6.586 0 Td[(1 an)]TJ/F20 7.97 Tf 6.587 0 Td[(1n;whereaiarethepositivesquarerootsoftheeigenvaluesofATA.Henceiffx=jxja)]TJ/F20 7.97 Tf 6.587 0 Td[(1x=jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(1x1;jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(1x2;:::;jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(1xn:Thenwecompute,f0ij=@ @xjfi=a)]TJ/F19 11.955 Tf 11.956 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(3xjxi+jxja)]TJ/F20 7.97 Tf 6.587 0 Td[(1ij:48

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Hence[f0f0T]ij=nXk=1[a)]TJ/F19 11.955 Tf 11.955 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.587 0 Td[(3xki+jxja)]TJ/F20 7.97 Tf 6.587 0 Td[(1ika)]TJ/F19 11.955 Tf 11.955 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(3xik+jxja)]TJ/F20 7.97 Tf 6.587 0 Td[(1ki];andthematrixhaseigenvaluesjajjxja)]TJ/F20 7.97 Tf 6.586 0 Td[(1;jxja)]TJ/F20 7.97 Tf 6.587 0 Td[(1;:::;jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(1| {z }n)]TJ/F20 7.97 Tf 6.587 0 Td[(1.Nowifjaj>1wehaveKIf=jaj;andKOf=jajn)]TJ/F20 7.97 Tf 6.586 0 Td[(1andifjaj<1wehaveKIf=jaj1)]TJ/F23 7.97 Tf 6.587 0 Td[(n;andKOf=jaj)]TJ/F20 7.97 Tf 6.587 0 Td[(1:UsingExample2.2.14thefollowingrelationshold,sincethemodulusthroughaarbi-trarypointiszero.KIf=KIf;KOf=KOf:WeillustratethisExamplebyconsideringthecasewhenn=3.Hence,fx=jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(1x=jxja)]TJ/F20 7.97 Tf 6.587 0 Td[(1x1;jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(1x2;jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(1x3=)]TJ/F19 11.955 Tf 5.48 -9.683 Td[(x21+x22+x231 2a)]TJ/F20 7.97 Tf 6.586 0 Td[(1x1;x21+x22+x231 2a)]TJ/F20 7.97 Tf 6.587 0 Td[(1x2;x21+x22+x231 2a)]TJ/F20 7.97 Tf 6.587 0 Td[(1x3:Thisimpliesthatf0x=a)]TJ/F19 11.955 Tf 11.956 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(3x21+jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(1a)]TJ/F19 11.955 Tf 11.956 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(3x2x1a)]TJ/F19 11.955 Tf 11.955 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(3x3x1a)]TJ/F19 11.955 Tf 11.955 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(3x1x2a)]TJ/F19 11.955 Tf 11.955 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(3x22+jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(1a)]TJ/F19 11.955 Tf 11.955 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(3x3x2a)]TJ/F19 11.955 Tf 11.955 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(3x1x3a)]TJ/F19 11.955 Tf 11.956 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(3x2x3a)]TJ/F19 11.955 Tf 11.955 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.587 0 Td[(3x23+jxja)]TJ/F20 7.97 Tf 6.587 0 Td[(1;49

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andthatf0xT=a)]TJ/F19 11.955 Tf 11.955 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.587 0 Td[(3x21+jxja)]TJ/F20 7.97 Tf 6.587 0 Td[(1a)]TJ/F19 11.955 Tf 11.955 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(3x1x2a)]TJ/F19 11.955 Tf 11.956 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(3x1x3a)]TJ/F19 11.955 Tf 11.955 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.587 0 Td[(3x2x1a)]TJ/F19 11.955 Tf 11.956 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(3x22+jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(1a)]TJ/F19 11.955 Tf 11.956 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(3x2x3a)]TJ/F19 11.955 Tf 11.955 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.587 0 Td[(3x3x1a)]TJ/F19 11.955 Tf 11.955 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(3x3x2a)]TJ/F19 11.955 Tf 11.955 0 Td[(1jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(3x23+jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(1:Withtheaidofasymboliccomputationprogramwendthepositivesquarerootsoftheeigenvaluesoff0f0Ttobejxja)]TJ/F20 7.97 Tf 6.586 0 Td[(1;jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(1;ajxja)]TJ/F20 7.97 Tf 6.587 0 Td[(3x21)-109(jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(3x21+ajxja)]TJ/F20 7.97 Tf 6.586 0 Td[(3x22)-109(jxja)]TJ/F20 7.97 Tf 6.587 0 Td[(3x22+ajxja)]TJ/F20 7.97 Tf 6.586 0 Td[(3x23)-109(jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(3x23+jxja)]TJ/F20 7.97 Tf 6.587 0 Td[(1:Thelasteigenvalueclearlysimpliestoajxja)]TJ/F20 7.97 Tf 6.586 0 Td[(1.Thereforetheimageoftheunitsphereunderthemapfisanellipsoidwithsemi-axisajxja)]TJ/F20 7.97 Tf 6.586 0 Td[(1;jxja)]TJ/F20 7.97 Tf 6.586 0 Td[(1;jxja)]TJ/F20 7.97 Tf 6.587 0 Td[(1.4.2FoldingExample4.2.1Letr;;zbethecylindricalcoordinatesofapointx;y;z2R3.Thisimpliesthatr0;02;z2R,withx=rcos;y=rsin;andz=z.Usingx;y;zwedenethemapC:R3!R3byCr;;z=x;y;z.ThedomainDdenedby0<<,iscalledawedgeofangle,0<2.WeconsidertwowedgesDandDwith.Deneahomeomorphismf:D!Dbyfr;;z=r; ;z;thismappingiscalledafolding.Wewishtocomputethevaluesofthesemi-axisofthedilatationellipsoida1;a2;a3.HenceweneedtondtheeigenvaluesofDFasdetailedbelow.x;y;zC)]TJ/F21 5.978 Tf 5.756 0 Td[(1)167(!r;;zF???yf???y^x;^y;^zC)]TJ/F19 11.955 Tf 29.224 0 Td[(r; ;z50

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WenotethatF=CfC)]TJ/F20 7.97 Tf 6.587 0 Td[(1.HenceDF=DCDfDC)]TJ/F20 7.97 Tf 6.586 0 Td[(1,whereDC)]TJ/F20 7.97 Tf 6.586 0 Td[(1=cos)]TJ/F19 11.955 Tf 11.291 0 Td[(sin0sincos0001;Df=1000 0001;andDC=cos sin 0)]TJ/F22 11.955 Tf 9.298 0 Td[(rsin rcos 0001:HencewiththeaidofasymboliccomputationprogramwendDF=coscos + sinsin r)]TJ/F20 7.97 Tf 6.587 0 Td[(1)]TJ/F25 11.955 Tf 5.479 -9.684 Td[()]TJ/F19 11.955 Tf 11.291 0 Td[(cos sin+ cossin 0r)]TJ/F23 7.97 Tf 6.814 -4.427 Td[( cos sin)]TJ/F19 11.955 Tf 11.956 0 Td[(cossin coscos +sinsin 0001:ThecharacteristicpolynomialofDFis)]TJ/F19 11.955 Tf 9.299 0 Td[(x)]TJ/F19 11.955 Tf 11.955 0 Td[(1 +x2)]TJ/F19 11.955 Tf 11.955 0 Td[(+ cos)]TJ/F22 11.955 Tf 13.15 8.088 Td[( x:Thereforewehavethefollowingeigenvalues:1;1 2+ cos)]TJ/F22 11.955 Tf 13.151 8.088 Td[( 1 2r + 2cos)]TJ/F22 11.955 Tf 13.151 8.088 Td[( 2)]TJ/F19 11.955 Tf 11.955 0 Td[(4 :Themaximumofthelargesteigenvalueoccurswhenweallowcos)]TJ/F23 7.97 Tf 13.679 5.256 Td[( tobe1,thereforeweobtaintheeigenvalues: ;1;1:51

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NowaccordingtoEquation.1.8wehaveKOf= andKIf= 2:4.3ConesExample4.3.1LetR;;bethesphericalcoordinatesofapointx;y;z2R3.ThisimpliesthatR0;0<2;0;withx=Rsincos;y=Rsinsin;andz=Rcos:Usingx;y;zwedenethemapS:R3!R3bySR;;=x;y;z.ThedomainCdenedby<,iscalledaconeofangle.When,deneahomeomorphismf:C!CbyfR;;=R;; .Wewishtocomputethevaluesofthesemi-axisofthedilatationellipsoida1;a2;a3.HenceweneedtondtheeigenvaluesofDFasdetailedbelow.x;y;zS)]TJ/F21 5.978 Tf 5.757 0 Td[(1)167(!R;;F???yf???y^x;^y;^zS)]TJ/F19 11.955 Tf 29.223 0 Td[(R;; WenotethatF=SfS)]TJ/F20 7.97 Tf 6.586 0 Td[(1.HenceDF=DSDfDS)]TJ/F20 7.97 Tf 6.586 0 Td[(1,whereDS)]TJ/F20 7.97 Tf 6.586 0 Td[(1=)]TJ/F19 11.955 Tf 11.291 0 Td[(cossinR)]TJ/F20 7.97 Tf 6.586 0 Td[(1sincsc)]TJ/F22 11.955 Tf 9.299 0 Td[(R)]TJ/F20 7.97 Tf 6.587 0 Td[(1coscos)]TJ/F19 11.955 Tf 11.291 0 Td[(sinsinR)]TJ/F20 7.97 Tf 6.587 0 Td[(1coscsc)]TJ/F22 11.955 Tf 9.299 0 Td[(R)]TJ/F20 7.97 Tf 6.587 0 Td[(1cossincos0)]TJ/F22 11.955 Tf 9.299 0 Td[(R)]TJ/F20 7.97 Tf 6.587 0 Td[(1sin;Df=10001000 ;52

PAGE 58

andDS=cossin sinsin cos )]TJ/F22 11.955 Tf 9.298 0 Td[(Rsinsin Rcossin 0Rcoscos Rcos sin)]TJ/F22 11.955 Tf 9.298 0 Td[(Rsin :HencewiththeaidofasymboliccomputationprogramwendDF= coscos +sinsin 0 cos sin+cossin 0cscsin 0Rcos sin)]TJ/F23 7.97 Tf 13.289 5.256 Td[( cossin 0coscos + sinsin :ThecharacteristicpolynomialofDFis[)]TJ/F22 11.955 Tf 9.299 0 Td[(x2+)]TJ/F22 11.955 Tf 13.276 8.088 Td[( cos)]TJ/F22 11.955 Tf 13.151 8.088 Td[( x)]TJ/F22 11.955 Tf 13.276 8.088 Td[( ]x)]TJ/F19 11.955 Tf 11.955 0 Td[(cscsin :Thereforewehavethefollowingeigenvalues:cscsin ;1 2[+ cos)]TJ/F22 11.955 Tf 13.151 8.088 Td[( r + 2cos2)]TJ/F22 11.955 Tf 13.151 8.088 Td[( )]TJ/F19 11.955 Tf 11.955 0 Td[(4 ]:Wenotethatforeach0x1,sinmx sinmisincreasinginm,when0m.Thusifx= ,wehavesin sinsin sin:.3.1Themaximumofthelasttwoeigenvalueswilloccurwhenweallowcos)]TJ/F23 7.97 Tf 13.188 5.256 Td[( =1,thereforeweobtaintheeigenvalues:cscsin ; ;1:WerecallfromCalculusthatsinx xisalwaysdecreasing.Henceforweobtaintherelationships sin sinand 1.NowaccordingtoEquation3.1.8andusing53

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Equation4.3.1,weobtainKIf=maxf2 2;sin2 sin2g;andKOf=2sin 2sin:54

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References[F]Fuglede,Bent.Extremallengthandfunctioncompletion;ActaMathematica1957,98,171-219.[HK]Homan,Kenneth;Kunze,Ray.LinearAlgebra;PrenticeHall:EnglewoodClis,NJ.1965.[IM]Iwaniec,Taduesz;Martin,Gavin.GeometricFunctionTheoryandNon-linearAnalysis;OxfordUniversityPress:NewYork,NY.2001.[L]Liouville,J.Extensionaucasdestroisdimensionsdelaquestiondutracegeographique;Applicationdel'analysealageometrie1850,609-616.[R]Rotman,Joseph.AdvancedModernAlgebra;PrenticeHall:UpperSaddleRiver,NJ.2002.[S]Spivak,Michael.CalculusonManifolds;W.ABenjaminInc.:NewYork,NY1965.[V]Vaisala,Jussi.Lecturesonn-dimensionalQuasiconformalMappings;LectureNotesinMathematics,Springer-Verlag,1971,229.[Z]Ziemer,William.WeaklyDierentialFunctions;Springer-Verlag:NewYork,NY.1989.55


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Analysis of quasiconformal maps in Rn
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In this thesis, we examine quasiconformal mappings in Rn. We begin by proving basic properties of the modulus of curve families. We then give the geometric, analytic,and metric space definitions of quasiconformal maps and show their equivalence. We conclude with several computational examples.
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Euclidean spaces.
Bounded distortion.
Moduli of curve families.
Dilation.
Absolute continuity on lines.
Jacobian.
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