Teichmuller Theory and the Universal Period Mappin

arXiv:alg-geom/9310005v28Oct1993TEICHM¨ULLERTHEORYANDTHEUNIVERSALPERIODMAPPINGVIAQUANTUMCALCULUSANDTHEH1/2SPACEONTHECIRCLEbySubhashisNagandDennisSullivanAbstract:Quasisymmetrichomeomorphismsofthecircle,thatariseintheTeichm¨ullertheoryofRiemannsurfacesasboundaryvaluesofquasiconfomaldiffeomorphismsofthedisk,havefractalgraphsingeneralandareconsequentlynotsoamenabletousualan-alyticalorcalculusprocedures.InthispaperwemakeuseoftheremarkablefactthisgroupQS(S1)actsbysubstitution(i.e.,pre-composition)asafamilyofboundedsymplec-ticoperatorsontheHilbertspaceH=“H1/2”(comprisingfunctionsmodconstantsonS1possessingasquare-integrablehalf-orderderivative).Conversely,andthatisalsoimpor-tantforourwork,quasisymmetrichomeomorphismsareactuallycharacterizedamongsthomeomorphismsofS1bythepropertyofpreservingthespaceH.InterpretingHviaboundaryvaluesasthesquare-integrablefirstcohomologyofthediskwiththecupproductsymplecticstructure,andcomplexstructureprovidedbytheHodgestar,weobtainauniversalformoftheclassicalperiodmappingextendingthemapof[12][13]fromDiff(S1)/Mobius(S1)toallofQS(S1)/Mobius(S1)–namelytotheentireuniversalTeichm¨ullerspace,T(1).ThetargetspacefortheperiodmapistheuniversalSiegelspaceofperiodmatrices;thatisthespaceofallthecomplexstructuresonHthatarecompatiblewiththecanonicalsymplecticstructure.UsingAlainConnes’suggestionofaquantumdifferentialdQJf=[J,f]–commutatorofthemultiplicationoperatorwiththecomplexstructureoperator–weobtaininlieuoftheproblematicalclassicalcalculusaquantumcalculusforquasisymmetrichomeomorphisms.Namely,onehasoperators{h,L},d◦{h,L},d◦{h,J},correspondingtothenon-linearclas-sicalobjectslog(h′),h′′h′dx,16Schwarzian(h)dx2definedwhenhisappropriatelysmooth.AnyoneoftheseobjectsisaquantummeasureoftheconformaldistortionofhinanalogywiththeclassicalcalculusBeltramicoefficientμforaquasiconformalhomeomorphismofthedisk.HereListhesmoothingoperatorontheline(orthecircle)withkernellog|x−y|,JistheHilberttransform(whichisd◦LorL◦d),and{h,A}meansAconjugatedbyhminusA.Theperiodmappingandthequantumcalculusarerelatedinseveralways.Forexam-ple,fbelongstoHifandonlyifthequantumdifferentialisHilbert-Schmidt.Also,thecomplexstructuresJonHlyingontheSchottkylocus(imageoftheperiodmap)satisfyaquantumintegrabilitycondition[dQJ,J]=0.Finally,wediscusstheTeichm¨ullerspaceoftheuniversalhyperboliclamination([20])asaseparablecomplexsubmanifoldofT(1).ThelatticeandK¨ahler(Weil-Petersson)metricaspectoftheclassicalperiodmappingappearbyfocusingattentiononthisspace.1§1-IntroductionTheUniversalTeichm¨ullerSpaceT(1),whichisauniversalparameterspaceforallRiemannsurfaces,isacomplexBanachmanifoldthatmaybedefinedasthehomoge-neousspaceQSS1
/M¨obS1
.HereQSS1
denotesthegroupofallquasisymmetrichomeomorphismsoftheunitcircleS1
,andM¨obS1
isthethree-parametersubgroupofM¨obiustransformationsoftheunitdisc(restrictedtotheboundarycircle).ThereisaremarkablehomogeneousK¨ahlercomplexmanifold,M=DiffS1
/M¨obS1
,–aris-ingfromapplyingtheKirillov-KostantcoadjointorbitmethodtotheC∞-diffeomorphismgroupDiffS1
ofthecircle([22])-thatclearlysitsembeddedinT(1)(sinceanysmoothdiffeomorphismisquasisymmetric).In[15]itwasprovedthatthecanonicalcomplex-analyticandK¨ahlerstructuresonthesetwospacescoincideunderthenaturalinjectionofMintoT(1).(TheK¨ahlerstructureonT(1)isformal–thepairingconvergesonpreciselytheH3/2vectorfieldsonthecircle.)Therelevantcomplex-analyticandsymplecticstructuresonM,(anditscloserelativeN=DiffS1
/S1
),arisefromtherepresentationtheoryofDiffS1
;whereasonT(1)thecomplexstructureisdictatedbyTeichm¨ullertheory,andthe(formal)K¨ahlermetricisWeil-Petersson.Thus,thehomogeneousspaceMisacomplexanalyticsubmanifold(notlocallyclosed)inT(1),carryingacanonicalK¨ahlermetric.Insubsequentwork([12][13])itwasshownthatonecancanonicallyassociateinfinite-dimensionalperiodmatricestothesmoothpointsMofT(1).Thecrucialstepinthisconstructionwasafaithfulrepresentation(Segal[18])ofDiffS1
ontheFrechetspaceV=C∞MapsS1,R
/R(theconstantmaps)(1)DiffS1
actsbypullbackonthefunctionsinVasagroupoftoplinearautomorphismsthatpreserveabasicsymplecticformthatVcarries.InordertobeabletoextendtheinfinitedimensionalperiodmaptothefullspaceT(1),itisnecessarytoreplaceVbyasuitable“completed”spacethatisinvariantunderquasisymmetricpullbacks.Moreover,thequasisymmetrichomeomorphismsshouldcon-tinuetoactasboundedsymplecticautomorphismsofthisextendedspace.ThesegoalsareachievedinthepresentpaperbydevelopingthetheoryoftheSobolevspaceonthecircleconsistingoffunctionswithhalf-orderderivative.ThisHilbertspaceH1/2=H,whichturnsouttobeexactlythecompletionofthepre-HilbertspaceV,actuallycharacterizesquasisymmetric(q.s).homeomorphisms(amongstallhomeomorphismsofS1).Thatfactwillbeimportantforourunderstandingoftheperiodmapping.Thesymplecticstructure,S,onVextendstoHandispreservedbytheactionofQS(S1),andindeedweshowthat2thisSistheuniquesymplecticstructureavailablewhichisinvariantundereventhetinyfinite-dimensionalsubgroupM¨obS1
(⊂QSS1
).WeutiliseseveraldifferentcharacterisationsofHanditscomplexification.Inpar-ticular,HcomprisesfunctionsonS1whicharedefinedquasi-every