Mean curvature 1 surfaces in hyperbolic 3-space wi

arXiv:math/0008015v1[math.DG]2Aug2000MEANCURVATURE1SURFACESINHYPERBOLIC3-SPACEWITHLOWTOTALCURVATUREIWAYNEROSSMAN,MASAAKIUMEHARA,ANDKOTAROYAMADADedicatedtoKatsuhiroShiohamaontheoccasionofhissixtiethbirthday.Abstract.Acompletesurfaceofconstantmeancurvature1(CMC-1)inhyperbolic3-spacewithconstantcurvature−1hastwonaturalnotionsof“totalcurvature”—oneisthetotalabsolutecurvaturewhichistheintegraloverthesurfaceoftheabsolutevalueoftheGaussiancurvature,andtheotheristhedualtotalabsolutecurvaturewhichisthetotalabsolutecurvatureofthedualCMC-1surface.Inthispaper,wecompletelyclassifyCMC-1surfaceswithdualtotalabsolutecurvatureatmost4π.Moreover,wegivenewexamplesandpartiallyclassifyCMC-1surfaceswithdualtotalabsolutecurvatureatmost8π.Withthedevelopmentsofthelastdecadeonconstantmeancurvature1(CMC-1)surfacesinhyperbolic3-spaceH3(thecompletesimply-connected3-manifoldofconstantsectionalcurvature−1),andwithsomanyexamplesnowknown,itisanaturalnextsteptoclassifyallsuchsurfaceswithlowtotalabsolutecurvature.AsCMC-1surfacesinH3sharequitesimilarpropertieswithminimalsurfacesinEuclidean3-spaceR3,letusfirstcommentthatthetotalabsolutecurvatureofaminimalsurfaceinR3isequaltothearea(countedwithmultiplicity)oftheGaussimageofthesurface,andthatcompleteminimalsurfacesinR3withtotalcurvatureatmost8πhavebeenclassified.(SeeLopez[6]andalsoTable2.)Furthermore,astheGaussmapofacompleteconformallyparametrizedminimalsurfaceisholomorphic,andhasawell-definedlimitateachendwhenthesurfacehasfinitetotalcurvature,theareaoftheGaussimagemustbeanintegermultipleof4π.ThequestionofclassifyinglowtotalcurvatureCMC-1surfacesinH3isanalogous—however,unlikeminimalsurfacesinR3,CMC-1surfacesinH3havetwoGaussmaps:thehyperbolicGaussmapGandthesecondaryGaussmapg.SotherearetwowaystoposethequestioninH3,withtwoverydifferentanswers.Onewayistoconsiderthetruetotalabsolutecurvature,whichistheareaoftheimageofg,butsincegmightnotbesingle-valuedonthesurface,thetotalcurvaturemightnotbeanintegermultipleof4π.ThisallowsformanymorepossibilitiesandmakestheproblemmoredifficultthanforminimalsurfacesinR3.Theauthorstakeupthisquestioninaseparatepaper[13].Thesecondway,whichisthethemeofthispaper,istostudytheareaoftheimageofG,whichwecallthedualtotalabsolutecurvature,asitisthetruetotalcurvatureofthedualCMC-1surface(whichwedefineinSection1)inH3.ThiswayhastheadvantagethatGissingle-valuedonthesurface,andsothedualtotalDate:July12,2000.2000MathematicsSubjectClassification.Primary53A10;Secondary53A35,53A42.12WAYNEROSSMAN,MASAAKIUMEHARA,ANDKOTAROYAMADAcurvatureisalwaysanintegermultipleof4π,likethecaseofminimalsurfacesinR3.Furthermore,thedualtotalcurvaturesatisfiesnotonlytheCohn-Vosseninequality,butalsothehyperbolicanalogueoftheOssermaninequality(whichcannotbesaidaboutthetruetotalcurvature)[19,23](seealso(2.1)inSection2).SothedualtotalcurvaturesharesmorepropertieswiththetotalcurvatureofminimalsurfacesinR3,motivatingourinterestinit.Inthispaper,weclassifyCMC-1surfaceswithdualtotalabsolutecurvatureatmost4π,andwegomuchofthewaytowardclassifyingCMC-1surfaceswithdualtotalabsolutecurvatureatmost8π(asafirststeptoafullclassificationofthe8πcase).InSection1,wegiveasummaryoftheresults,andinSection2wegivepreliminariesforthelattersections.TheclassificationofCMC-1surfaceswithdualtotalabsolutecurvaturelessthanorequalto4πisgiveninSection3.Surfaceswithdualtotalabsolutecurvature8πarediscussedinSection4—andtherewefindnewexamples,weclassifycertaincases,andweshownonexistenceincertainothercases.InSection5,fromdeformationsofcorrespondingminimalsurfacesinR3,weproducetwoclassesofnewCMC-1surfaceswithdualtotalabsolutecurvature8π.Forthereaders’convenience,weattachAppendixAtoexplainthecomputationoflog-termcoefficientsofsecondorderlinearordinarydifferentialequationswithregularsingularities.1.SummaryoftheresultsTostateourresultsprecisely,webeginwithsomenotations.Letf:M→H3beacompleteconformalCMC-1immersionofaRiemannsurfaceMintoH3.ByBryant’srepresentationformula,thereisaholomorphicnullimmersionF:fM→SL(2,C)suchthatf=FF∗,wherefMistheuniversalcoverofMandF∗=tF.(“null”meansdet(F−1dF)=0.)Here,weconsiderH3=SL(2,C)/SU(2)={aa∗|a∈SL(2,C)}[1,15].WecallFtheliftoff,andFsatisfiesdF=Fg−g21−gQdg(1.1)onfM,whereg(thesecondaryGaussmap)isameromorphicfunctiondefinedonfMandQ(theHopfdifferential)isaholomorphic2-differentialonM.Thentheinducedmetricds2andcomplexificationofthesecondfundamentalformhareds2=(1+|g|2)2Qdg2,h=−Q−Q+ds2.By(1.1),thesecondaryGaussmapsatisfiesg=−dF12dF11=−dF22dF21,whereF(z)=F11(z)F12(z)F21(z)F22(z).ThemapgisdetermineduniquelyuptoaM¨obiustransformationg7→a⋆gbya∈SU(2),where,forgenerala=(aij)∈SL(2,C),wedenotea⋆g:=a11g+a12a21g+a22.ThehyperbolicGaussmapGoffisdefinedbyG=dF11dF21=dF12dF22,CMC-1SURFACESOFLOWTOTALCURVATUREI3whichcanbeinterpretedasstereographicprojectionoftheendpointsinthesphereatinfinityofH3oftheorientednormalgeodesicsemanatingfromthesurface.Inparticular,GisameromorphicfunctiononM.TheinversematrixF−1isalsoaholomorphicnullimmersion,andproducesanewCMC-1immersionf#=F−1(F−1)∗:fM→H3,calledthedualoff[19].Theinducedmetricds2#andtheHopfdifferentialQ#off#areds2#=(1+|G|2