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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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]
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]