On the biharmonic and harmonic indices of the Hopf

arXiv:math/0402295v1[math.DG]18Feb2004ONTHEBIHARMONICANDHARMONICINDICESOFTHEHOPFMAPE.LOUBEAUANDC.ONICIUCAbstract.Biharmonicmapsarethecriticalpointsofthebienergyfunctionaland,fromthispointofview,generaliseharmonicmaps.WeconsidertheHopfmapψ:S3→S2andmodifyitintoanonharmonicbiharmonicmapφ:S3→S3.Weshowφtobeunstableandestimateitsbiharmonicindexandnullity.ResolvingthespectrumoftheverticalLaplacianassociatedtotheHopfmap,werecoverUrakawa’sdeterminationofitsharmonicindexandnullity.1.IntroductionInordertoanalysethespaceofsmoothmapsbetweenmanifolds,EellsandSamp-sonintroducedin[7]thefamilyoffunctionals:Ek(φ)=12ZM|(d+d⋆)kφ|2vg,k∈N⋆formapsφ:(M,g)→(N,h).Theyquicklyspecialisedtheirstudytok=1andcalledthecriticalpointsoftheenergyE1harmonicmaps,thecorrespondingEuler-Lagrangeequationbeingthevanishingofthetensionfieldτ(φ)=trace∇φdφ.Here∇φdenotestheconnectioninthepull-backbundleφ−1TN.Recently,thecaseofthebienergyE2hasbeenthesubjectofsomescrutiny.ItscriticalpointsarecalledbiharmonicmapsandJiang,in[11],obtainedthefirstandsecondvariationformulas,showing,inparticular,thatamapisbiharmonicifandonlyifτ2(φ)=0,whereτ2(φ)=−Jφ(τ(φ))andJφistheJacobioperatorofthesecondvariationfortheenergy.AsJφislinear,anyharmonicmapisbiharmonic,soweareinterestedinnonharmonicbiharmonicmaps.Naturally,biharmonicsubmanifoldshavebeenthefirstcentreofattention.Onsur-facesofrevolution,Caddeo,MontaldoandPiuobtainedtheparametricequationofallnongeodesicbiharmoniccurvesonthetorusandDelaunaysurfaces[3],nonmini-malbiharmonicsubmanifoldsofS3areclassifiedin[4],whileconstructionsofsuchsubmanifoldsonSn(n≥4)arepresentedin[5].Inthepresenceofnonconstantsectionalcurvature,aparametricdescriptionofallnongeodesicbiharmoniccurvesoftheHeisenberggroupisgivenin[6],whereasIn-oguchiclassified,in[10],thebiharmonicLegendrecurvesandHopfcylindersona3-dimensionalSasakianspaceformandSasaharagivesin[18]theexplicitrepre-sentationofnonminimalbiharmonicLegendresurfacesina5-dimensionalSasakianspace.Whilethisresultsdemonstratetheexistenceofnontrivialbiharmonicmaps,though1991MathematicsSubjectClassification.58E20,31B30.Keywordsandphrases.Harmonicandbiharmonicmaps,Riemanniansubmersions,stability.TheauthorsaregratefultoT.Levasseurforhishelpwithrepresentationtheory.ThesecondauthorthankstheC.N.R.S.foragrantwhichmadepossibleathree-monthstayattheUniversit´edeBretagneOccidentaleinBrest.12E.LOUBEAUANDC.ONICIUCnotinabundance,theycarrylittleornoindicationastotheirglobalbehaviour.Tothiseffect,someefforthasbeendirectedtowardsthestabilityofbiharmonicmaps.Asanexample,theidentitymapSn→SnandthetotallygeodesicinclusionmapSm→Snarebiharmonicstableandtheirnullitieswerecomputedin[15].In[19],theauthorprovedthatthenonminimalbiharmonicLegendresurfacesina5-dimensionalSasakianspaceareunstable.TheindexoftheinclusionSn(1√2)→Sn+1wascomputedin[13],anditisex-actlyone.Then,wereinvestigatedtheindicesofbiharmonicmapsintheunitEuclideansphereSn+1obtainedfromminimalRiemannianimmersionsinSn(1√2).Inparticular,theauthorsshowedthattheindexofthenonminimalbiharmonicmapSmqm+1m
→Sm+p+1derivedfromthegeneralisedVeronesemap,isatleast2m+3.Thepresentarticlepursuesthesamelineofresearchforsubmersions,bymodifyingaharmonicRiemanniansubmersionψ:M→Sn(1√2)intoanonharmonicbiharmonicsubimmersionφ:M→Sn+1whichweprovetobeunstable.ConcentratingontheHopfmapψ:S3(√2)→S2(1√2),wefirstestablishthespectrumoftheverticalLaplacianofψwhichallowsustodeterminetheharmonicindexandnullityofψ,hencerecoveringUrakawa’sresult(cf.[21]).Wealsogiveageometriccharacter-izationofkerJψ,andobtainanestimateofthebiharmonicindexandnullityofφ:S3(√2)→S3:theindexisatleast11andthenullityisgreaterthan8.Throughoutthepaper,manifolds,metrics,mapsareassumedtobesmooth,and(M,g)denotesaconnectedRiemannianmanifoldwithoutboundary.Wedenoteby∇theLevi-Civitaconnectionof(M,g),andforthecurvaturetensorfieldRweusethesignconventionR(X,Y)=[∇X,∇Y]−∇[X,Y],whiletheLaplacianonsectionsofφ−1TNisΔφ=−trace(∇φ)2.LetSn(r√2)=Sn(r√2)×{r√2}={p=(x1,...,xn+1,r√2)(x1)2+···+(xn+1)2=r22}beahypersphereofSn+1(r),asprovedin[4],Sn(r√2)isanonminimalbiharmonicsubmanifoldofSn+1(r).Indeed,letη=1r(x1,...,xn+1,−r√2)beaunitsectionofthenormalbundleofSn(r√2)inSn+1(r).Then,thesecondfundamentalformofSn(r√2)isB(X,Y)=∇di(X,Y)=−1rhX,Yiηandthetensionfieldoftheinclusionmapi:Sn(r√2)→Sn+1(r)isτ(i)=−nrη.Besides,bydirectcomputation,τ2(i)=0.Theorem1.1.LetMbeacompactmanifoldandψ:M→Sn(r√2)anonconstantmap.Themapφ=i◦ψ:M→Sn+1(r)isnonharmonicbiharmonicifandonlyifψisharmonicande(ψ)isconstant.Proof.Thecompositionlawgivesτ(φ)=τ(ψ)−2re(ψ)η.Bystraightforwardcalculations:Δφτ(φ)=Δψτ(ψ)+2r2e(ψ)τ(ψ)+1r2dφ(θ♯+4grade(ψ))−1r4r2(e(ψ))2+2Δe(ψ)−2divθ♯+|τ(ψ)|2
η,andtraceRSn+1(r)(dφ·,τ(φ))dφ·=1r2dφ(θ♯)−2r2e(ψ)τ(ψ)+4r3(e(ψ))2η,ONTHEBIHARMONICANDHARMONICINDICESOFTHEHOPFMAP3whereθisa1-formonMgivenbyθ(X)=hdψ(X),τ(ψ)i,andθ♯∈C(TM)isdefinedbyhθ♯,Xi=θ(X),X∈C(TM).Replacinginthebiharmonicequationweobtain:τ2(φ)=−Δψτ(ψ)−2r2dφ(θ♯+2grade(ψ))+1r2Δe(ψ)−2divθ♯+|τ(ψ)|2
η.Assumethatφisbiharmonic.Astheη-partofτ2(φ)vanishes:2Δe(ψ)−2divθ♯+|τ(ψ)|2=0.ByStokes,wededucethatτ(ψ)=0whichimpliesΔe(ψ)=0,i.e.e(ψ)iscons