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

ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 30, Number 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 obtain a variable reduction principle for the Willmore variational problem in an ample class of conformal structures on S2n+1. This variational problem is transformed into another one, associated with an elastic-energy functional with potential, on spaces of curves in CPn.Then, we give a simple method to construct Willmore tori in certain conformal structures on S2n+1. Moreover, we exhibit some families of Willmore tori for the standard conformal class on S3and S7. 1. Introduction. Let S2n+1 be the unit sphere in Cn+1 endowed with the standard metric ¯g. The unit circle S1acts naturally on S2n+1 to produce CPnas orbit space. The canonical projection π:(S2n+1,¯g)→(CPn,g) is a Riemannian submersion, where g denotes the Fubini-study metric of constant holomorphic sectional curvature 4. A vertical, unit global vector field Vis defined on S2n+1 by V(z)=iz, for all z∈S2n+1. The horizontal distribution His defined to be the ¯g-orthogonal complementary to the orbits. As usual, overbars will denote horizontal lifts of the corresponding objects in a Riemannian submersion (see [6], [13] for details about notation and terminology). In particular, the Levi-Civita connections ¯ ∇and ∇of ¯g and g, respectively, are related via the following well-known formulae: ¯ ∇¯ X¯ Y=∇XY−¯g(i¯ X, ¯ Y)V,(1.1) ¯ ∇¯ XV=¯ ∇V¯ X=i¯ X,(1.2) ¯ ∇VV=0.(1.3) Remark 1. (i) It should be noticed that the last formula shows the geodesic nature of the orbits in (S2n+1,¯g). (ii) Since πmay also be Received by the editors on November 24, 1998. 1991 AMS Mathematics Subject Classification. 53C40, 53A05. Key words and phrases. Willmore torus, Kaluza-Klein metric, conformal structure, φ-elastic curve. Copyright c 2000 Rocky Mountain Mathematics Consortium 815 816 J.L. CABRERIZO AND M. FERN´ ANDEZ regarded as the projection of a principal fiber bundle with structure group S1and His S1-invariant, it defines a principal connection whose connection 1-form will be denoted by ω. (iii) We can use the nice argument of Pinkall (see [15]), to show that an immersed surface M in S2n+1 is S1-invariant if and only if M=Mγ=π−1(γ)forsome immersed curve γin CPn. In particular, if γis closed, then Mγis a torus,whichisembeddedifγis free of self-intersections in CPn. Let hbe a Riemannian metric on CPnand ua positive smooth function on CPn. We define (1.4) ¯ hu=π∗(h)+ε(u◦π)2ω∗(dt2), where dt2is the usual metric on S1and ε=±1. It is clear that ¯ hu is a metric on S2n+1, which is Riemannian or Lorentzian according to whether εis +1 or −1, respectively. These metrics are called the generalized Kaluza-Klein metrics on S2n+1 ([9]). It is not difficult to see that the S1-action on S2n+1 is made up through isometries of (S2n+1,¯ hu). Furthermore, π:(S2n+1,¯ hu)→ (CPn,h) is a pseudo-Riemannian submersion, which has geodesic fibers if and only if uis constant. In this case the scalar curvature of (S2n+1,¯ hu) is constant. Moreover, if γis a curve with curvature function kin (CPn,h), then the mean curvature function αof Mγ in (S2n+1,¯ hu)satisfies[1]: (1.5) α2=1 4(k2◦π). Let Nbe the space of immersions of a genus one compact surface N in S2n+1. For any semi-Riemannian metric  hon S2n+1,wehavethe Willmore functional W:N→Rdefined by (1.6) W(ϕ)=N (α2+S)dv, where αis the mean curvature function of ϕ,Sis the sectional curvature function of (S2n+1, h)alongϕand dv isthevolumeelementofϕ∗( h) on N. The critical points of this functional are the so-called Willmore tori. This functional is an invariant under conformal changes of the WILLMORE TORI 817 ambient metric  h([7]). Therefore, if C(¯ hu) denotes the conformal class associated to ¯ hu, it is natural to pose the following problem: Studying the existence and characterization of S1-invariant Willmore tori in (S2n+1,C(¯ hu)). Some particular answers to this problem have been obtained in [1], [5], [15]. On the other hand, we consider the total squared curvature functional acting on closed curves (or curves satisfying given first order boundary data) in a Riemannian manifold (M,g). The extremal points of this functional are called free elastic curves in (M,g)(see[10], [11], [12]). In this note we show that the existence of Willmore tori in (S2n+1,C(¯ hu)) which are invariant under the natural S1-action on S2n+1 is equivalent to the existence of critical points of the functional (1.7) F(γ)=γ (k2+φ(γ)) ds, acting on closed curves γin (CPn,(1/u2)h), where kis the curvature function of γand φ(γ)=4ε(g(γ,γ ))2worksasapotential. A φ-elastic curve is a critical point of (1.7). Then, we will use the EulerLagrange equation associated with the functional (1.7), to construct Willmore tori in a wide family of conformal structures on S2n+1 (see Corollary 3.1). In particular, we obtain families of Willmore tori in (S3,C(¯g)) and (S7,C(¯g)) (see Corollaries 3.2 and 3.3). 2. The main theorem. Theorem 2.1. Mγ=π−1(γ)is a Willmore torus in (S2n+1,C(¯ hu)) if and only if γisaclosedcurveinCPn, which is a critical point of the following elastic-energy functional on (CPn,(1/u2)h): (2.1) F(γ)=γ (k2+φ(γ)) ds, where kis the curvature function of γand φ(γ)=4ε(g(γ,γ))2. Proof. Since the Willmore variational problem is invariant under conformal changes of the ambient space metric, we choose the following 818 J.L. CABRERIZO AND M. FERN´ ANDEZ metric in C(¯ hu): (2.2)  hu=1 (u◦π)2¯ hu=π∗1 u2h+εω∗(dt2). This choice has the following advantage: π:(S2n+1, hu)→(CPn,(1/ u2)h) has geodesic fibers. It is clear that the Willmore functional is S1-invariant, that is, W(eiθϕ)=W(ϕ). We define the submanifold NS1of S1-invariant immersions which can be identified (see (iii) of Remark 1) with Mγ= {π−1(γ)|γis a closed curve immersed in CPn}.LetΣbetheset of critical points of W(Willmore tori), and denote by ΣS1the set of critical points of Wwhen restricted to NS1. Then we use the principle of symmetric criticality ([14]) to get (2.3) Σ ∩N S1=Σ S1. Therefore, to obtain Willmore tori in (S2n+1,C(¯ hu)) which do not break the S1-symmetry of the problem, we only need to compute Won NS1 and then to proceed in due course. To compute W(π−1(γ)), we parametrize γby its arc length in (CPn,(1/u2)h) and observe that Tp(Mγ)isamixedsectionofTp(S2n+1) for any p∈Mγ.Since hu(V,V )=ε,thetermSin the integrand of W is given by ([6]): (2.4) S=ε hu( Du ¯ XV,  Du ¯ XV), where ¯ Xis the horizontal lift of X=γand  Duis the Levi-Civita connection of  hu. Take a local horizontal frame {¯ X,i ¯ X,Y2,iY 2,... ,Y n,iY n} along Mγ.Thenweuse(1.2)toget  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 calculate the Lie brackets appearing in the last formula, we use (1.1) and then (2.5)  Du ¯ XV=εg(γ,γ)i¯ X, WILLMORE TORI 819 gbeing the Fubini-study metric in CPn. Using (1.5), (2.3) and (2.4), we have W(π−1(γ)) = L 02π 01 4κ2+ε(g(γ,γ))2ds dt =π 2L 0 (κ2+φ(γ)) ds, where Lis the length of γin (CPn,(1/u2)h)andφ(γ)=4ε(g(γ,γ))2. This completes the proof of the theorem. In particular, if n=1,weidentifyCP1with S2in the standard fashion to obtain the usual Hopf map π:S3→S2. On the other hand, as a consequence of the uniformization theorem for Riemann surfaces, we can choose in the conformal class of ¯ huametric (2.6) ¯gu=π∗(g)+ε(u◦π)2ω∗(dt2), where gis the canonical metric of constant Gaussian curvature 4 in S2. So we have Corollary 2.2. Let γbeaclosedimmersedcurveinS2.Then Mγ=π−1(γ)is a Willmore torus in (S3,C(¯gu)) if and only if γis a φ-elastica with potential φ(γ)=4ε(g(γ,γ))2in (S2,(1/u2)g). 3. Further discussions and applications. Let γbe a φ-elastica in (CPn,(1/u2)h). The potential φis a smooth function, defined on the unit tangent vector bundle of (CPn,(1/u2)h). It is clear that φis a basic function on that bundle, i.e., a function on CPnif and only if his chosen in the conformal class of the Fubini-study metric gon CPn. In this case, without loss of generality, we can take h=g. For basic potentials, the Euler-Lagrange equations of φ-elasticae can be computed using Lemma 1.1 of [10] in a standard argument which involves some integration by parts. Then we have (3.1) 2 ∇3 TT+3 ∇T(κ2T)+2 R( ∇TT,T)T+ ∇φ−φ ∇TT−T(φ)T=0, where the elements appearing in this formula are taken in (CPn,(1/u2)h), in particular  Ris the Riemann curvature of this metric and φ=4u4. 820 J.L. CABRERIZO AND M. FERN´ ANDEZ Let γbe a closed curve in CPnand denote by ηits unit normal vector field in (CPn,g). Put Fγ +to name the space of positive smooth functions, f,onCPnsuch that η(f) = 0 (along γ). We have Corollary 3.1. Let γbe a geodesic in (CPn,g)and u∈F γ +.Then Mγis a Willmore torus in (S2n+1,C(¯gu)) which is conformally minimal in (S2n+1,¯gu). Proof. A direct computation shows that (3.1) can be written as (3.2) 2 ∇3 T+3 ∇T(κ2T)+2 R( ∇TT,T)T−φ3∇T∗T∗=0, where T∗is the unit tangent of γcomputed in (CPn,g). Now it is obvious that if γis a geodesic in (CPn,g)andu∈F γ +,thenitisalsoa geodesic in (CPn,(1/u2)g)andsoaφ-elastica in (CPn,(1/u2)g)with φ=4u4. Now the statement follows from the main theorem. Corollary 3.2. Let γbe any great circle in (S2,g)and u∈F γ +. Then Mγ=π−1(γ)is a Willmore torus in (S3,C(¯gu)). It is obvious that minimal surfaces of the standard sphere (Sm,¯g)are Willmore. If we pay attention to the spectral behavior of the position vector of those surfaces in Rm+1 [16], then it seems natural to look for Willmore surfaces in (Sm,¯g) which can be constructed in Rm+1 using eigenfunctions of the Laplacian coming from exactly two different eigenvalues (2-type surfaces [8]). These surfaces have been completely classified in [3]. They are certain flat tori which fully yield in (S5,¯g)or in (S7,¯g). Since the family of Willmore tori in (S5,¯g) has been studied in [5], in this note we are going to deal with those surfaces in (S7,¯g). It was shown in [3] that the map Y:R2→C4,givenby (3.3) Y(s, t)=eit(c1cos As, c1sin As, c2cos Bs, c2sin Bs), with c2 1+c2 2= 1, defines an isometric immersion of a flat torus, say T,in (S7,¯g) which is of 2-type when A=Band Willmore for certain choices of (A, B) which involve the isometry type of that flat torus (see [3]). It is evident that these Willmore tori are S1-invariant. Namely, if we put γ(s)=π(c1cos As, c1sin As, c2cos Bs,c2sin Bs), then T=π−1(γ). WILLMORE TORI 821 Furthermore, one can prove that γ(s) is a helix which yields into a three-dimensional, Lagrangian and totally geodesic RP3of CP3.It should be noticed that, according to our main theorem, the curves γ(s) are closed helices, which are φ-elasticae (in this case elasticae, [10]) with φ= 4 (because of its constancy, φworks as a Lagrange multiplier, [10]). Now we get Willmore tori in (S7,C(¯g)) by lifting closed helices, which are elasticae (with φ=4)in(CP3,g). We can use a similar argument to that used in [2]and[4] to obtain a one-parameter family of elastic helices, φ=4,inRP3. In particular, this family contains a rational one-parameter subfamily of closed elastic helices. Now we regard RP3as a Lagrangian and totally geodesic submanifold in CP3 to obtain, via our main theorem, the following family of Willmore tori in (S7,C(¯g)) which includes those of 2-type given in (3.3). Corollary 3.3. 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O’Neill, Semi-Riemannian geometry, Academic Press, New York, 1983. 14. R.S. Palais, Critical point theory and the minimax principle,inGlobal Analysis, Proc. Sympos. Pure Math. 15 (1970), 185 212. 15. U. Pinkall, Hopf tori in S3, Invent. Math. 81 (1985), 379 386. 16. T. Takahasi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380 385. Departamento de Geometr  a y Topolog  a, Facultad de Matem  aticas, Universidad de Sevilla, Apdo. Correos 1160, 41080 Sevilla, Spain E-mail address: [email protected] Departamento de Geometr  a y Topolog  a, Facultad de Matem  aticas, Universidad de Sevilla, Apdo. Correos 1160, 41080 Sevilla, Spain E-mail address: [email protected]