A characterization of Weingarten surfaces in hyper

arXiv:0709.2441v1[math.DG]15Sep2007ACHARACTERIZATIONOFWEINGARTENSURFACESINHYPERBOLIC3-SPACENIKOSGEORGIOUANDBRENDANGUILFOYLEAbstract.Westudy2-dimensionalsubmanifoldsofthespaceL(H3)ofori-entedgeodesicsofhyperbolic3-space,endowedwiththecanonicalneutralK¨ahlerstructure.SuchasurfaceisLagrangianiffthereexistsasurfaceinH3orthogonaltothegeodesicsofΣ.WeprovethattheinducedmetriconaLagrangiansurfaceinL(H3)haszeroGausscurvatureifftheorthogonalsurfacesinH3areWeingarten:theeigenvaluesofthesecondfundamentalformarefunctionallyrelated.WethenclassifythetotallynullsurfacesinL(H3)andrecoverthewell-knownholomor-phicconstructionsofflatandCMC1surfacesinH3.RecentlytheexistenceanduniquenessofacanonicalneutralK¨ahlerstructureonthespaceL(H3)oforientedgeodesicsofhyperbolic3-spaceH3hasbeenestablished[3][10].ThemainpurposeofthispaperistoapplythisworktothestudyofsurfacesSinH3.TheorientedgeodesicsnormaltoSformasurfaceinL(H3)whichisLagrangianwithrespecttothisK¨ahlerstructure.Infact,thestudyofsurfacesinH3isequivalent,atleastlocally,tothestudyofLagrangiansurfacesinL(H3).Specialclassesofsurfacesinhyperbolic3-spacehavebeenstudiedformanydecades,withvariousconstructionsbeingdevelopedandappliedtotheparticularclassofsurfacesunderconsideration.Forexample,surfacesofconstantmeancur-vature1havebeeninvestigatedin[1],whileflatsurfaceshavebeentreatedin[5][6][9].ThecommonfeatureofthesesurfacesisthattheyareWeingarten[12]:theyhavesomespecifiedfunctionalrelationshipbetweentheeigenvaluesofthesecondfundamentalform.Otherformsofthisrelationshiphavebeenconsidered[7]anduniquenessresultsobtainedforWeingartensurfacesinR3[2][4].InthispaperwegiveanewcharacterizationoftheWeingartenconditionforsurfacesinH3:MainTheorem:LetS⊂H3beaC4smoothorientedsurfaceandΣ⊂L(H3)betheLagrangiansurfaceformedbytheorientedgeodesicsnormaltoS.ThenSisWeingartenifftheGausscurvatureoftheLorentzmetricinducedonΣbytheneutralK¨ahlermetriciszero.Theproofofthisresultfollowsfromacarefulstudyof2-dimensionalsubmani-foldsofL(H3)usinglocalcoordinates,movingframesandthecorrespondencespace.Date:15thSeptember,2007.1991MathematicsSubjectClassification.Primary:51M09;Secondary:51M30.Keywordsandphrases.Kaehlerstructure,hyperbolic3-space,Weingartensurfaces.12NIKOSGEORGIOUANDBRENDANGUILFOYLEInthefollowingsectionwegivethegeometricbackground(furtherdetailscanbefoundin[3])andexplorethesubmanifoldtheoryofsurfacesinL(H3).SectionthreecontainstheproofoftheMainTheorem.InthefinalsectionweclassifytheholomorphicLagrangiansurfacesinL(H3)andrecovertheholomorphicconstructionsofflatandCMC1surfacesinH3.1.GeometricBackgroundWebrieflyrecallthebasicconstructionofthecanonicalneutralK¨ahlermetriconthespaceL(H3)oforientedgeodesicsofH3-furtherdetailscanbefoundin[3].WeuseoneoftwomodelsofH3,thePoincar´eballmodel:B3={(y1,y2,y3)∈R3|(y1)2+(y2)2+(y3)21},withstandardcoordinates(y1,y2,y3)onR3,andhyperbolicmetricds2=4[(dy1)2+(dy2)2+(dy3)2][1−(y1)2−(y2)2−(y3)2]2.andtheupper-halfspacemodel:R3+={(x0,x1,x2)∈R3|x00},forstandardcoordinates(x0,x1,x2)onR3.Inthesecoordinatesthehyperbolicmetrichasexpression:d˜s2=(dx0)2+(dx1)2+(dx2)2(x0)2.ThesearerelatedbythemappingR3+→B3:(x0,x1,x2)7→(y1,y2,y3)definedbyy1=2x1(x0+1)2+(x1)2+(x2)2,y2=2x2(x0+1)2+(x1)2+(x2)2,y3=(x0)2+(x1)2+(x2)2−1(x0+1)2+(x1)2+(x2)2.AnorientedgeodesicinH3isuniquelydeterminedbyitsbeginningandendpointontheboundaryoftheballmodel,andsoL(H3)canbeidentifiedwithS2×S2−Δ,whereΔisthediagonalinS2×S2.EndowingS2×S2−Δwiththestandarddifferentiablestructure,atangentvectortoanorientedgeodesicγ∈L(H3)canthenbeidentifiedwithanorthogonalJacobifieldalongγ⊂H3.RotationofJacobifieldsthrough900aboutγdefinesanalmostcomplexstructureonL(H3).Thisalmostcomplexstructureisintegrable,andsoL(H3)becomesacomplexsurface,whichturnsouttobebiholomorphictoP1×P1−Δ.HereΔisthe“reflected”diagonal:intermsofholomorphiccoordinates(μ1,μ2)onP1×P1,Δ={(μ1,μ2):μ1¯μ2=−1}.ThisdistinctionbetweenP1×P1−ΔandP1×P1−Δiscrucial,asexplainedinsection4.2.ThecomplexstructureJonL(H3)canbesupplementedwithacompatiblesym-plecticstructureΩ,whichhasthefollowingexpressioninholomorphiccoordinates:Ω=−1(1+μ1¯μ2)2dμ1∧d¯μ2+1(1+¯μ1μ2)2d¯μ1∧dμ2.(1.1)WEINGARTENSURFACESINHYPERBOLIC3-SPACE3TogetherweobtainaK¨ahlermetricG(·,·)=Ω(J·,·):G=−i1(1+μ1¯μ2)2dμ1⊗d¯μ2−1(1+¯μ1μ2)2d¯μ1⊗dμ2.(1.2)Thismetric,whichhassignature++−−,isinvariantundertheactioninducedonL(H3)bytheisometrygroupofH3.Indeed,thishasbeenshowntobetheuniqueK¨ahlermetriconL(H3)withthisproperty[10].InordertotransfergeometricdatabetweenL(H3)andH3weuseacorrespon-dencespace:π1L(H3)×R@@@@RΦL(H3)?H3Thekeypropertyofthiscorrespondenceisthat,givenγ∈L(H3),thesetΦ◦π−11(γ)istheorientedgeodesicinH3,while,forapointp∈H3,π1◦Φ−1(p)isthesetoforientedgeodesicsinL(H3)thatpassthroughp.ThemapΦtakesanorientedgeodesicγinL(H3)andarealnumberrtothepointonγanaffineparameterdistancerfromsomefixedpointonthegeodesic.Thischoiceofpointoneachgeodesiccanbemadeglobally,butwemoreoftenjustusealocalchoice,whichissufficientforourpurposes.Intermsofholomorphiccoordinates(μ1,μ2)onL(H3)andupper-halfspacecoordinates(x0,x1,x2)themapΦhasexpression:z=1−μ1¯μ22¯μ2+1+μ1¯μ22¯μ2tanhr,t=|1+¯μ1μ