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App1.Math.J.ChineseUniv

    ;2008,23(1):6568

    ;NewconstructionofspecialLagrangiansubmanifoldsin

    ;Rnequippedwiththestandardmetric

    ;HANYingbo

    ;Abstract.AclassoftwistedspecialLagrangiansubmanifoldsinTRandakindofaustere ;submanifoldfromconormalbundleofminimalsurfaceofR0areconstructed. ;?1Introduction

    ;SpecialLagrangiansumbmanifoldsmavbedefinedasthosesubmanifoldswhichareboth ;Lagrangianandminima1.Alternatively,theyarecharacterisedasthosesubmnaifoldswhichare

    ;calibratedbyacertainnform.sotheyhavetheremarkablepropertyofbeingareaminimizing.

    ;Thesesubmanifoldshavereceivedmanyattentionrecentlyduetoconnectionwithstringtheory.

    ;Moreparticularly,understandingfibrationofspecialLagrangianinCalabi’ylaumanifoldsof

    ;dimension3iscrucialformirrorsymmetry.ExamplesofspecialLagrangiansubmanifolds ;areveryimportantforstudyingthesesubmanifolds.Inrecentyears,someexamplesofthese ;submanifoldshavebeenconstructedbymanypeople.Forexample.HarveyandLawson[jJgave

    ;someexamplesofspecialLagrangiansumbmanifoldsinC.andespeciallytheyconstructeda

    ;classofspecialLagrangiansubmanifoldsbyusingnormalbundles.Joycein[2-5

    gaveexplicit

    ;examplesofspecialLagrangiansubmanifoldsinC.Borisenko[6Jconstructedalotoftwisted

    ;specialLagrangiansubmanifoldsinTR3fromthetwistednormalbundleofminimalsurfaces

    ;giveninR3.Thisisageneralizationofnormalbundlegivenin

    11withdimension3.Bryant[71

    ;gaveaclassoftwistedspecialLagrangiansubmanifoldsinC3.whicharedifierentfromtheones

    ;Austereisalsoaspecialkindofminimalsubmanifold,whichwerefirstgivenin11.Inrecent

    ;years,therearesomepeoplewhostudythesesubmanifolds.Forexample,BrantIsj,Dajczerand

    ;Florit[gavemanypropertiesofthesesubmanifolds.

    ;HarvayandLawson[1Jgavesomespecial

    ;Lagrangiansubmanifoldbyusingausteresubmanifolds.

    ;InthispaperweconstructakindoftwistedspecialLagrangiansubmanifoldsinTRfrom ;thetwistednormalbundleofminimalsurfacesinR,andthisisageneralizationofthecase ;giveninl6linhigherdimension.WealsofindakindofausteresubmanifoldinC0=R6,and ;thengetalotofspecialLagrangiansubmanifclldsinTR6.

    ;?2Preliminaries

    ;Webeginbydefiningcalibrationandcalibratedsubmanifolds,followingHarvayandLawson[

    ;MathematicsClassification:53D12.

    ;Keywords:twistedspecialLagrangiansubmanifold,twistednormalbundle,austeresubmanifold

    ;DigitalObjectIdentifier(DOI):10.1007/sl176600801090.

    ;SupportedbytheZhongdiangrantofNSFC.

    ;

    ;App1.Math.3.ChineseUnivV_01.23,No.1

    ;Definition2.1.Let(M,)beaRiemannianmanifold.AnorientedtangentplaneVonM

    ;isavectorsubspaceVofsometangentspaceMtoMwithdimV=k,equippedwithan ;orientation.IfVisanorientedtangentplaneonMthenIisaEuclideanmetriconV,so

    ;combiningglvwiththeorientationonVgivesanaturalvolumeformvolvonV,whichisa ;kformonV.

    ;NowletbeaclosedformonM.WesaythatisacalibrationonMifforeveryoriented ;

    ;planeVonMwehavelvolv.Herel=avolvforsomeo/ER,andIvolv

    ;ifo/1.LetNbeanorientedsubmanifoldofMwithdimensionk.Theneachtangent ;spaceNforENisanorientedtangent后一plane.W_esayNisacalibratedsubmani-m),(2.1) ;=

    ;dz1A…Adz,(2.2)

    ;thenReandImarerea1m-formsonC.LetLbeanorientedrea1submanifoldofCof ;realdimensionm,letE[0,27r).WesaythatLisaspecialLagrangiansubmanifoldofC, ;withe,ifLiscalibratedwithrespecttocosRe+sin9Im~2,inthesenseofDefinition2.1. ;Usuallywetake=0,sothatLhasphase1,andiscalibratedwithrespecttoRe. ;HarveyandLawson[]gavethefollowingalternativecharacterizationofspecialLagrangian

    ;submanfolds.

    ;Lemma2.3.LetLbearea1m-dimensiona1submanifoldofC.ThenLadmitsanorientation ;makingitintoaspecialLagrangiansubmanifoldofCwithphaseeifandonlyiflL=0 ;andfsin0Ref~COSOIm~)lL=0.

    ;Definition2.4.LetXbeadimensionalsubmanifoldofR,andAthesecondfundamental ;formofX.ThenAliesinSXforeachEX.WecallXaustereifallEXand

    ;Y?,theinvariantsofoddorderofquadraticformA.YinSXvanish.Equivalently,for ;allxEXandYE,thecollectionofeigenvaluesAl,…,ofA.Y,withmultiplicity,should

    ;beinvariantundermultiplicationby1.

    ;Lemma2.5.(see[1)LetXbeadimensionalsubmanifoldofR.Thenthenormalbundle ;()={,Y):E;YE=X}isspecialLagrangiansubmanifoldinTR=C,

    ;withphasei.ifandonlyifXisaustere.

    ;Borienko[6]whousedtwistednormalbundlesgaveakindofspecialLagrangiansubmanifolds

    ;inC3.

    ;Lemma2.6.(see[6)LetXbeanoriented,regularminimalsurfaceinR0andP:X__?Ra ;harmonicfunction.Let(8,t)belocalcoordinatesonXcompatiblewiththeorientation,and ;writetheimmersionX__?R0as(8,t)__?X(s,t).Definevector-valuedfunctionsn,P:X__?R

    ;bv

    ;n=

    ;i,p=×n,(2.3)n,p1=×n,’J

    ;whereX8=ox,

    ;andsoon.Thenn,Parewell-definedandindependentofchoiceofcoordinates ;(8,t).Define

    ;N={+t(p()+rn(x)):EX,rER},

    ;thenNisaruledspecialLagrangiansubmanifoldinC.withphasei. ;

    ;HAN]ZingboNewconstructionofspecialLagrangiansubmanifoldsinTRequippedwith?.?67

    ;?3MainresultsfortwistedspecialLagrangiansubmanifolds ;Theorem3?1?LetMbeanorientedminimalsurfaceofR,andP:M}Rbeasmooth

    ;functiononM.LetNbe

    ;N={(,v+(vp):v??(M)),x?M),

    ;where(M)isanormalspaceofMinR.ThenNisaspecialLagrangiansubmanifoldin ;R=Cwithphasei(n--2)ifandonlyifPisaharmonicfunctiononM. ;Proof.Letel,c2,c3,…,Cn)beanorthonormalframefieldofR,el,e2}aretangentto

    ;M.AndNcanbeexpressed38

    ;n

    ;:(8,t)((s),>ez+(vp)),for8=(81,s2),t(t3,…,t)

    ;/=3

    ;wherex=(Xl…,xn)?McR.

    ;Thetangentspacetothisembeddingat(s,t)isspannedbythevectors ;E1=(el,A(e1)+.(p)),E2=(e2,A(e2)+V..(p)),Ea=(0,e3),…,=(0,en),

    ;whereAisthesecondfundamentalformofMintoR.Nextwedividetheproofintotwosteps. ;Step(I1NisaLagrangiansubmanifoldinTR.

    ;WenowconsiderthecomplexstructureJdefinedonTR=CbysettingJ(X.Y1= ;(),whereX,YarevectorsofR.Let(,)bethecannoicalmetriconC. ;(,Ez)=(ei,e1)=0fori=1,2f=3,…,n

    ;(JE1,)一一(A(e1),e2)(.(vp),e2)+(A(e2),e1)+(V..(Vp),e1)

    ;=

    ;((e1),e2)Hess(p)(el,e2)+(A(e2),e1)+Hess(p)(e2,e1)=0.

    ;ForAissymmetricoperatoronM,andHess(p)isasymmetricmatrix,thenweknowthat ;NisaLagrangiansubmanifoldinTR=C.

    ;Step(II)NisaspecialLagrangiansubmanifoldinTR.

    ;WedenotethestandardcoordinatesinTR=C=R0R”by(Xl,?

;andsetzi=xi+~/=iunderthestandardorthonormalbasis{el,…,en)

    ;(n,0)form/2=dZlA…AdznistheholomorphicvolumeformofTR

    ;knowthatRe/2isacalibratedformonC.

    ;,

    ;

    ;,n,1,…,n)

    ;ofR.Thenthe

    ;C.Itiseasyto

    ;FromLemma2.3,weknowthatNisaspecialLagrangiansubmanifoldwithrespecttothe

    ;calibratedformRe((T)2--n/2)ofcifandonlyif

    ;Im((v/Ti-)2--n)(E1,E2,E3,…,)=0

    ;Inthenextpart,wecomputeIm((v/Tf)2--n9)(E1,E2,Ea,…,).

    ;(~/=_)2--n/2(El,E2,Ea,…,)

    ;=

    ;()2--ndz1Adz2Adz3A…Adz(E1,E2,Ea,.

    ;Fromtheequation

    ;(1+(1+(.(vp),e1)))(1+(2+(..(vp),e2)))

    ;(3.1)weknow

    ;Im((v/Ti-)2--n)(E1,E2,Ea,…,)

    ;=1+2+(.(vp),e1))+(v..(vp),e2))

    ;=trA”+Ap.

    ;SinceMisaminimalsurfaceofR.thentrA”=0.Sowehave

    ;Im((,//1)2--nO)(E1,E2,Ea,…,En)=Ap

    ;ThusweknowthatNisaspecialLagrangiansubmanifoldinTR=Cwithphasei()if

    ;andonlyifAp=0,i.e.,PisaharmonicfunctiononM. ;

    ;App1.Math.J.ChineseUnivV01.23.No.1

    ;Remark3.2.Whenn=3,theSalTleresultswerefoundin6,sothistheoremisageneraliza- ;tionoftheresultsof[61.

    ;Remark3.3.WhenPisaconstantfunctiononM,thetwistednormalbundleisthecanniocal

    ;normalbundleofMinTRngivenin[1].

    ;Lemma3.4.Let:MRbeadimensionalsubmanifoldofR,thenwehavethat: ;?X=kH.

    ;whereHisthemeancurvaturevectorofM.IfMisaminimalsubmanifoldin,soxiisa

    ;harmonicfunctiononM,whereX=(Xl,…,xn).

    ;FromLemma3.4wecangetalotofharmonicfunctionsonM,andmanyspecialLagrangian

    ;submanifoldsinTB=Caswel1.

    ;Theorem3.5.Leti:R0beaminimalsurfaceofR.,theconormalbundleofMisgiven

    ;byN:{(z,tn):x?M,t?R),andnistheunitnormalvectorofMinR.,thenNisan ;austeresubmanifoldinTR0.

    ;Furthermore,weknowthatL=N(N)isaspecialLagrangian ;submnaifoldinTR0=C0.

    ;Proof.ByassumptionweknowthatNisa3-dimensionalspecialLagrangiansubmanifold

    ;ofR0:C0,thenNisaminimalsubmanifoldofC0.Itiseasytoknowthat(0,tn)is

;ageodesiclineinbothNandC.,thus(0,n)isaprinciplecurvaturevectorofA”,andthe

    ;Drinciplecurvaturecorrespondingto(0,n)iszero.ThenweknowNisanausteresubmanifold

    ;ofC0.wherevisanysectionofnormalbundleofNinC0andAisthesecondfundamentalform ;of?inCa.FromLemma2.5weknowthatL=N(N)isaspecialLagrangiansubmanifold ;ofR0:C6,whereL=N(N)istheconormalbundleofNinTR=C.

    ;Remark3.6.FromTheorem3.5wecanconstructalotofspecialLagrangiansubmanifolds ;inC6.

    ;Acknowledgements.TheauthorwouldliketothankProfessorDongYXf0rhisencourage- ;mentandsupport.

    ;Rferences

    ;1HarveyR,LawsonHB.Calibratedgeometries,ActaMath,1982,148:47-157. ;2JoyceDD.SpecialLagrangianm-foldsinCwithsymmetries,MathDG/0008021,2000? ;3JoyceDD.ConstructingspecialLagrangianmfoldsinCbyevolvingquadrics,MathD

    G/0008155,

    ;2000.

    ;4JoyceDD.EvolutionequationsforspecialLagrangian3-foldsinC,MathDG/0010036,2000.

    ;5JoyceDD.RuledspecialLagrangian3-foldsinC.,MathDG/0012060,2000. ;6BorisenkoA.RuledspecialLagrangiansurfaces,In:ATFomenkoed.,MinimalSurface,AM

    ;S,1993,15:269-285.

    ;7BrantRL.SecondfamiliesofspecialLagrangian3-folds,MathDG/0007128,2000. ;8BryantRL.SomeremarksonthegeometryofAusteremanifolds,BolSocBrasilMat,1991,21:

    ;33一】57.

    ;9DajczerM,FloritLA.AclassofAusteresubmanifolds,IllinoisJournalofMathematics,2001

    ;45(3):735755.

    ;Dept.ofMath.,SoutheastUniv.,Nanjing211189,China.

    ;

    ;

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