Willmore Tori in a wide family of conformal structures on odd dimensional spheres

ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 30, Numbe 3, Fall 2000 WILLMORE TORI IN A WIDE FAMILY OF CONFORMAL STRUCTURES ON ODD DIMENSIONAL SPHERES J.L. CABRERIZO AND M. FERN´ ANDEZ ABSTRACT. We ob ain a a iable educ ion p inciple o he Willmo e a ia ional p oblem in an ample class o con o - mal s uc u es on S2n+1. This a ia ional p oblem is ans- o med in o ano he one, associa ed wi h an elas ic-ene gy unc ional wi h po en ial, on spaces o cu es in CPn.Then, we gi e a simple me hod o cons uc Willmo e o i in ce ain con o mal s uc u es on S2n+1. Mo eo e , we exhibi some amilies o Willmo e o i o he s anda d con o mal class on S3and S7. 1. In oduc ion. Le S2n+1 be he uni sphe e in Cn+1 endowed wi h he s anda d me ic ¯g. The uni ci cle S1ac s na u ally on S2n+1 o p oduce CPnas o bi space. The canonical p ojec ion π:(S2n+1,¯g)→(CPn,g) is a Riemannian subme sion, whe e g deno es he Fubini-s udy me ic o cons an holomo phic sec ional cu a u e 4. A e ical, uni global ec o field Vis defined on S2n+1 by V(z)=iz, o all z∈S2n+1. The ho izon al dis ibu ion His defined o be he ¯g-o hogonal complemen a y o he o bi s. As usual, o e ba s will deno e ho izon al li s o he co esponding objec s in a Riemannian subme sion (see [6], [13] o de ails abou no a ion and e minology). In pa icula , he Le i-Ci i a connec ions ¯ ∇and ∇o ¯g and g, espec i ely, a e ela ed ia he ollowing well-known o mulae: ¯ ∇¯ X¯ Y=∇XY−¯g(i¯ X, ¯ Y)V,(1.1) ¯ ∇¯ XV=¯ ∇V¯ X=i¯ X,(1.2) ¯ ∇VV=0.(1.3) Rema k 1. (i) I should be no iced ha he las o mula shows he geodesic na u e o he o bi s in (S2n+1,¯g). (ii) Since πmay also be Recei ed by he edi o s on No embe 24, 1998. 1991 AMS Ma hema ics Subjec Classi ica ion. 53C40, 53A05. Key wo ds and ph ases. Willmo e o us, Kaluza-Klein me ic, con o mal s uc- u e, φ-elas ic cu e. Copy igh c 2000 Rocky Moun ain Ma hema ics Conso ium 815 816 J.L. CABRERIZO AND M. FERN´ ANDEZ ega ded as he p ojec ion o a p incipal fibe bundle wi h s uc u e g oup S1and His S1-in a ian , i defines a p incipal connec ion whose connec ion 1- o m will be deno ed by ω. (iii) We can use he nice a gumen o Pinkall (see [15]), o show ha an imme sed su ace M in S2n+1 is S1-in a ian i and only i M=Mγ=π−1(γ) o some imme sed cu e γin CPn. In pa icula , i γis closed, hen Mγis a o us,whichisembeddedi γis ee o sel -in e sec ions in CPn. Le hbe a Riemannian me ic on CPnand ua posi i e smoo h unc ion on CPn. We define (1.4) ¯ hu=π∗(h)+ε(u◦π)2ω∗(d 2), whe e d 2is he usual me ic on S1and ε=±1. I is clea ha ¯ hu is a me ic on S2n+1, which is Riemannian o Lo en zian acco ding o whe he εis +1 o −1, espec i ely. These me ics a e called he gene alized Kaluza-Klein me ics on S2n+1 ([9]). I is no difficul o see ha he S1-ac ion on S2n+1 is made up h ough isome ies o (S2n+1,¯ hu). Fu he mo e, π:(S2n+1,¯ hu)→ (CPn,h) is a pseudo-Riemannian subme sion, which has geodesic fibe s i and only i uis cons an . In his case he scala cu a u e o (S2n+1,¯ hu) is cons an . Mo eo e , i γis a cu e wi h cu a u e unc ion kin (CPn,h), hen he mean cu a u e unc ion αo Mγ in (S2n+1,¯ hu)sa isfies[1]: (1.5) α2=1 4(k2◦π). Le Nbe he space o imme sions o a genus one compac su ace N in S2n+1. Fo any semi-Riemannian me ic  hon S2n+1,weha e he Willmo e unc ional W:N→Rdefined by (1.6) W(ϕ)=N (α2+S)d , whe e αis he mean cu a u e unc ion o ϕ,Sis he sec ional cu a u e unc ion o (S2n+1, h)alongϕand d is he olumeelemen o ϕ∗( h) on N. The c i ical poin s o his unc ional a e he so-called Willmo e o i. This unc ional is an in a ian unde con o mal changes o he WILLMORE TORI 817 ambien me ic  h([7]). The e o e, i C(¯ hu) deno es he con o mal class associa ed o ¯ hu, i is na u al o pose he ollowing p oblem: S udying he exis ence and cha ac e iza ion o S1-in a ian Willmo e o i in (S2n+1,C(¯ hu)). Some pa icula answe s o his p oblem ha e been ob ained in [1], [5], [15]. On he o he hand, we conside he o al squa ed cu a u e unc ional ac ing on closed cu es (o cu es sa is ying gi en fi s o de bounda y da a) in a Riemannian mani old (M,g). The ex emal poin s o his unc ional a e called ee elas ic cu es in (M,g)(see[10], [11], [12]). In his no e we show ha he exis ence o Willmo e o i in (S2n+1,C(¯ hu)) which a e in a ian unde he na u al S1-ac ion on S2n+1 is equi alen o he exis ence o c i ical poin s o he unc ional (1.7) F(γ)=γ (k2+φ(γ)) ds, ac ing on closed cu es γin (CPn,(1/u2)h), whe e kis he cu a u e unc ion o γand φ(γ)=4ε(g(γ,γ ))2wo ksasapo en ial. A φ-elas ic cu e is a c i ical poin o (1.7). Then, we will use he Eule - Lag ange equa ion associa ed wi h he unc ional (1.7), o cons uc Willmo e o i in a wide amily o con o mal s uc u es on S2n+1 (see Co olla y 3.1). In pa icula , we ob ain amilies o Willmo e o i in (S3,C(¯g)) and (S7,C(¯g)) (see Co olla ies 3.2 and 3.3). 2. The main heo em. Theo em 2.1. Mγ=π−1(γ)is a Willmo e o us in (S2n+1,C(¯ hu)) i and only i γisaclosedcu einCPn, which is a c i ical poin o he ollowing elas ic-ene gy unc ional on (CPn,(1/u2)h): (2.1) F(γ)=γ (k2+φ(γ)) ds, whe e kis he cu a u e unc ion o γand φ(γ)=4ε(g(γ,γ))2. P oo . Since he Willmo e a ia ional p oblem is in a ian unde con o mal changes o he ambien space me ic, we choose he ollowing 818 J.L. CABRERIZO AND M. FERN´ ANDEZ me ic in C(¯ hu): (2.2)  hu=1 (u◦π)2¯ hu=π∗1 u2h+εω∗(d 2). This choice has he ollowing ad an age: π:(S2n+1, hu)→(CPn,(1/ u2)h) has geodesic fibe s. I is clea ha he Willmo e unc ional is S1-in a ian , ha is, W(eiθϕ)=W(ϕ). We define he submani old NS1o S1-in a ian imme sions which can be iden ified (see (iii) o Rema k 1) wi h Mγ= {π−1(γ)|γis a closed cu e imme sed in CPn}.Le Σbe hese o c i ical poin s o W(Willmo e o i), and deno e by ΣS1 he se o c i ical poin s o Wwhen es ic ed o NS1. Then we use he p inciple o symme ic c i icali y ([14]) o ge (2.3) Σ ∩N S1=Σ S1. The e o e, o ob ain Willmo e o i in (S2n+1,C(¯ hu)) which do no b eak he S1-symme y o he p oblem, we only need o compu e Won NS1 and hen o p oceed in due cou se. To compu e W(π−1(γ)), we pa ame ize γby i s a c leng h in (CPn,(1/u2)h) and obse e ha Tp(Mγ)isamixedsec iono Tp(S2n+1) o any p∈Mγ.Since hu(V,V )=ε, he e mSin he in eg and o W is gi en by ([6]): (2.4) S=ε hu( Du ¯ XV,  Du ¯ XV), whe e ¯ Xis he ho izon al li o X=γand  Duis he Le i-Ci i a con- nec ion o  hu. Take a local ho izon al ame {¯ X,i ¯ X,Y2,iY 2,... ,Y n,iY n} along Mγ.Thenweuse(1.2) oge  Du ¯ XV=−1 2 hu([ ¯ X,i ¯ X],V)i¯ X−1 2 hu([ ¯ X,Yj],V)Yj −1 2 hu([ ¯ X,iYj],V)iYj. To calcula e he Lie b acke s appea ing in he las o mula, we use (1.1) and hen (2.5)  Du ¯ XV=εg(γ,γ)i¯ X, WILLMORE TORI 819 gbeing he Fubini-s udy me ic in CPn. Using (1.5), (2.3) and (2.4), we ha e W(π−1(γ)) = L 02π 01 4κ2+ε(g(γ,γ))2ds d =π 2L 0 (κ2+φ(γ)) ds, whe e Lis he leng h o γin (CPn,(1/u2)h)andφ(γ)=4ε(g(γ,γ))2. This comple es he p oo o he heo em. In pa icula , i n=1,weiden i yCP1wi h S2in he s anda d ashion o ob ain he usual Hop map π:S3→S2. On he o he hand, as a consequence o he uni o miza ion heo em o Riemann su aces, we can choose in he con o mal class o ¯ huame ic (2.6) ¯gu=π∗(g)+ε(u◦π)2ω∗(d 2), whe e gis he canonical me ic o cons an Gaussian cu a u e 4 in S2. So we ha e Co olla y 2.2. Le γbeaclosedimme sedcu einS2.Then Mγ=π−1(γ)is a Willmo e o us in (S3,C(¯gu)) i and only i γis a φ-elas ica wi h po en ial φ(γ)=4ε(g(γ,γ))2in (S2,(1/u2)g). 3. Fu he discussions and applica ions. Le γbe a φ-elas ica in (CPn,(1/u2)h). The po en ial φis a smoo h unc ion, defined on he uni angen ec o bundle o (CPn,(1/u2)h). I is clea ha φis a basic unc ion on ha bundle, i.e., a unc ion on CPni and only i his chosen in he con o mal class o he Fubini-s udy me ic gon CPn. In his case, wi hou loss o gene ali y, we can ake h=g. Fo basic po en ials, he Eule -Lag ange equa ions o φ-elas icae can be compu ed using Lemma 1.1 o [10] in a s anda d a gumen which in ol es some in eg a ion by pa s. Then we ha e (3.1) 2 ∇3 TT+3 ∇T(κ2T)+2 R( ∇TT,T)T+ ∇φ−φ ∇TT−T(φ)T=0, whe e he elemen s appea ing in his o mula a e aken in (CPn,(1/u2)h), in pa icula  Ris he Riemann cu a u e o his me ic and φ=4u4. 820 J.L. CABRERIZO AND M. FERN´ ANDEZ Le γbe a closed cu e in CPnand deno e by ηi s uni no mal ec o field in (CPn,g). Pu Fγ + o name he space o posi i e smoo h unc ions, ,onCPnsuch ha η( ) = 0 (along γ). We ha e Co olla y 3.1. Le γbe a geodesic in (CPn,g)and u∈F γ +.Then Mγis a Willmo e o us in (S2n+1,C(¯gu)) which is con o mally minimal in (S2n+1,¯gu). P oo . A di ec compu a ion shows ha (3.1) can be w i en as (3.2) 2 ∇3 T+3 ∇T(κ2T)+2 R( ∇TT,T)T−φ3∇T∗T∗=0, whe e T∗is he uni angen o γcompu ed in (CPn,g). Now i is ob ious ha i γis a geodesic in (CPn,g)andu∈F γ +, heni isalsoa geodesic in (CPn,(1/u2)g)andsoaφ-elas ica in (CPn,(1/u2)g)wi h φ=4u4. Now he s a emen ollows om he main heo em. Co olla y 3.2. Le γbe any g ea ci cle in (S2,g)and u∈F γ +. Then Mγ=π−1(γ)is a Willmo e o us in (S3,C(¯gu)). I is ob ious ha minimal su aces o he s anda d sphe e (Sm,¯g)a e Willmo e. I we pay a en ion o he spec al beha io o he posi ion ec o o hose su aces in Rm+1 [16], hen i seems na u al o look o Willmo e su aces in (Sm,¯g) which can be cons uc ed in Rm+1 using eigen unc ions o he Laplacian coming om exac ly wo diffe en eigen alues (2- ype su aces [8]). These su aces ha e been comple ely classified in [3]. They a e ce ain fla o i which ully yield in (S5,¯g)o in (S7,¯g). Since he amily o Willmo e o i in (S5,¯g) has been s udied in [5], in his no e we a e going o deal wi h hose su aces in (S7,¯g). I was shown in [3] ha he map Y:R2→C4,gi enby (3.3) Y(s, )=ei (c1cos As, c1sin As, c2cos Bs, c2sin Bs), wi h c2 1+c2 2= 1, defines an isome ic imme sion o a fla o us, say T,in (S7,¯g) which is o 2- ype when A=Band Willmo e o ce ain choices o (A, B) which in ol e he isome y ype o ha fla o us (see [3]). I is e iden ha hese Willmo e o i a e S1-in a ian . Namely, i we pu γ(s)=π(c1cos As, c1sin As, c2cos Bs,c2sin Bs), hen T=π−1(γ). WILLMORE TORI 821 Fu he mo e, one can p o e ha γ(s) is a helix which yields in o a h ee-dimensional, Lag angian and o ally geodesic RP3o CP3.I should be no iced ha , acco ding o ou main heo em, he cu es γ(s) a e closed helices, which a e φ-elas icae (in his case elas icae, [10]) wi h φ= 4 (because o i s cons ancy, φwo ks as a Lag ange mul iplie , [10]). Now we ge Willmo e o i in (S7,C(¯g)) by li ing closed helices, which a e elas icae (wi h φ=4)in(CP3,g). We can use a simila a gumen o ha used in [2]and[4] o ob ain a one-pa ame e amily o elas ic helices, φ=4,inRP3. In pa icula , his amily con ains a a ional one-pa ame e sub amily o closed elas ic helices. Now we ega d RP3as a Lag angian and o ally geodesic submani old in CP3 o ob ain, ia ou main heo em, he ollowing amily o Willmo e o i in (S7,C(¯g)) which includes hose o 2- ype gi en in (3.3). Co olla y 3.3. A a ional one-pa ame e amily o Willmo e o i exis s in (S7,C(¯g)) which ha e nonze o cons an mean cu a u e in (S7,¯g). Mo eo e , hey yield ully in he se en sphe e. Rema k 2. I should be no iced ha he amily o Willmo e o i gi en in his co olla y is diffe en om ha ob ained in [3]. Howe e , bo h amilies ha e a nonemp y in e sec ion made up by imme sions defined in (3.3). Acknowledgmen s. The au ho s would like o since ely hank M. Ba os o his aluable commen s and sugges ions. 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Co eos 1160, 41080 Se illa, Spain E-mail add ess: [email p o ec ed] Depa amen o de Geome  a y Topolog  a, Facul ad de Ma em  a icas, Uni e sidad de Se illa, Apdo. Co eos 1160, 41080 Se illa, Spain E-mail add ess: [email p o ec ed]