Mean curvature 1 surfaces in hyperbolic 3-space wi

MEANCURVATURE1SURFACESINHYPERBOLIC3-SPACEWITHLOWTOTALCURVATUREIIWAYNEROSSMAN,MASAAKIUMEHARA,ANDKOTAROYAMADADedicatedtoProfessorKatsueiKenmotsuonhissixtiethbirthdayAbstract.Inthiswork,completeconstantmeancurvature1(CMC-1)surfacesinhyperbolic3-spacewithtotalabsolutecurvatureatmost4¼areclassified.Thisclassifi-cationsuggeststhattheCohn-Vosseninequalitycanbesharpenedforsurfaceswithoddnumbersofends,andaproofofthisisgiven.1.IntroductionThisisacontinuation(PartII)ofthepaper[14](PartI)withthesametitle.AspointedoutinPartI,completeCMC-1(constantmeancurvature1)surfacesfinthehyperbolic3-spaceH3havetwoimportantinvariants.OneisthetotalabsolutecurvatureTA(f),andtheotheristhedualtotalabsolutecurvatureTA(f#),whichisthetotalabsolutecurvatureofthedualsurfacef#.InPartI,weinvestigatedsurfaceswithlowTA(f#).HereweinvestigateCMC-1surfaceswithlowTA(f).ClassifyingCMC-1surfacesinH3withlowTA(f)ismoredifficultthanclassifyingthosewithlowTA(f#),forthefollowingreasons:TA(f)equalstheareaofthesphericalimageofthe(holomorphic)secondaryGaussmapg,andgmightnotbesingle-valuedonthesurface.Therefore,TA(f)isgenerallynota4¼-multipleofaninteger,unlikethecaseofTA(f#).Furthermore,theOssermaninequalitydoesnotholdforTA(f),alsounlikethecaseofTA(f#).TheweakerCohn-VosseninequalityisthebestgenerallowerboundforTA(f)(withequalityneverholding[19]).InSection3,weshallprovethefollowing:Theorem1.1.Letf:M2!H3beacompleteCMC-1immersionoftotalabsolutecurvatureTA(f)·4¼.Thenfiseither(1)ahorosphere,(2)anEnnepercousin,(3)anembeddedcatenoidcousin,(4)afinite±-foldcoveringofanembeddedcatenoidcousinwithM2=Cnf0gandsecondaryGaussmapg=z¹for¹·1=±,or(5)awarpedcatenoidcousinwithinjectivesecondaryGaussmap.Thehorosphereistheonlyflat(andconsequentlytotallyumbilic)CMC-1surfaceinH3.ThecatenoidcousinsaretheonlyCMC-1surfacesofrevolution[3].TheEnneper2000MathematicsSubjectClassification.Primary53A10;Secondary53A35,53A42.12WAYNEROSSMAN,MASAAKIUMEHARA,ANDKOTAROYAMADAcousinsareisometrictominimalEnnepersurfaces[3].Thewarpedcatenoidcousins[19]arelesswell-knownandaredescribedinSection2.Althoughthistheoremissimplystated,forthereasonsstatedabovetheproofismoredelicatethanitwouldbeiftheconditionTA(f)·4¼werereplacedwithTA(f#)·4¼,orifminimalsurfacesinR3withTA·4¼wereconsidered.CMC-1surfacesfwithTA(f#)·4¼areshowninPartItobeonlyhorospheres,Ennepercousinduals,catenoidcousins,andwarpedcatenoidcousinswithembeddedends.Itiswell-knownthattheonlycompleteminimalsurfacesinR3withTA·4¼aretheplane,theEnnepersurface,andthecatenoid.Weseefromthistheoremthatanythree-endedsurfacefsatisfiesTA(f)4¼,andsotheCohn-Vosseninequalityisnotsharpforsuchf.Ontheotherhand,theCohn-Vosseninequalityissharpforcatenoidcousins,andanumericalexperimentin[15]showsittobesharpforgenus0surfaceswith4ends.Thisraisesthequestion:WhichclassesofsurfacesfhaveastrongerlowerboundforTA(f)thanthatgivenbytheCohn-Vosseninequality?Pursuingthis,inSection4weshowthatstrongerlowerboundsexistforgenuszeroCMC-1surfaceswithanoddnumberofends.WeextendTheorem1.1inafollow-upwork[15],tofindaninclusivelistofpossibilitiesforCMC-1surfaceswithTA(f)·8¼,andconsiderwhichpossibilitieswecanclassifyorfindexamplesfor.(MinimalsurfacesinR3withTA·8¼areclassifiedbyLopez[9].ThosewithTA·4¼arelistedinTable1.)2.PreliminariesLetf:M!H3beaconformalCMC-1immersionofaRiemannsurfaceMintoH3.Letds2,dAandKdenotetheinducedmetric,inducedareaelementandGaussiancurvature,respectively.ThenK·0andd¾2:=(¡K)ds2isaconformalpseudometricofconstantcurvature1onM.Wecallthedevelopingmapg:fM:=(theuniversalcoverofM)!CP1thesecondaryGaussmapoff,whereCP1isthecomplexprojectiveline.Namely,gisaconformalmapsothatitspull-backoftheFubini-StudymetricofCP1equalsd¾2:(2.1)d¾2=(¡K)ds2=4dgd¯g(1+g¯g)2:Bydefinition,thesecondaryGaussmapgoftheimmersionfisuniquelydetermineduptotransformationsoftheform(2.2)g7!a?g:=a11g+a12a21g+a22a=Ãa11a12a21a22!2SU(2):Inadditiontog,twootherholomorphicinvariantsGandQarecloselyrelatedtogeometricpropertiesofCMC-1surfaces.ThehyperbolicGaussmapG:M!CP1isholomorphicandisdefinedgeometricallybyidentifyingtheidealboundaryofH3withCMC-1SURFACESOFLOWTOTALCURVATUREII3CP1:G(p)istheasymptoticclassofthenormalgeodesicoff(M)startingatf(p)andorientedinthemeancurvaturevector’sdirection.TheHopfdifferentialQisthesymmetricholomorphic2-differentialonMsuchthat¡Qisthe(2;0)-partofthecomplexifiedsecondfundamentalform.TheGaussequationimplies(2.3)ds2¢d¾2=4Q¢Q;where¢meansthesymmetricproduct.Moreover,theseinvariantsarerelatedby(2.4)S(g)¡S(G)=2Q;whereS(¢)denotestheSchwarzianderivativeS(h):=µh00h0¶0¡12µh00h0¶2#dz2µ0=ddz¶withrespecttoacomplexcoordinatezonM.SinceK·0,wecandefinethetotalabsolutecurvatureasTA(f):=ZM(¡K)dA2[0;+1]:ThenTA(f)istheareaoftheimageinCP1ofthesecondaryGaussmap.TA(f)isgenerallynotanintegermultipleof4¼—forcatenoidcousins[3,Example2]andtheir±-foldcovers,TA(f)admitsanypositiverealnumber.ForeachconformalCMC-1immersionf:M!H3,thereisaholomorphicnullimmer-sionF:fM!SL(2;C),theliftoff,satisfyingthedifferentialequation(2.5)dF=FÃg¡g21¡g!!;!=Qdgsuchthatf=FF¤,whereF¤=tF.HereweconsiderH3=SL(2;C)=SU(2)=faa¤ja2SL(2;C)g.IfF=(Fij),equation(2.5)impliesg=¡dF12dF