Hamiltonian minimal Lagrangian spheres in the product of spheres

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1 Symposium Valenciennes 43 Hamiltonian minimal Lagrangian spheres in the product of spheres ILDEFONSO CASTRO Departamento de Matemáticas, Universidad de Jaén, Jaén, Spain Abstract We find out the only Hamiltonian minimal Lagrangian sphere in the non-einstein Kähler manifold S 2 (r 1 ) S 2 (r 2 ), product of two spheres of different radii r 1 r 2. Keywords : Hamiltonian minimal Lagrangian surfaces, Willmore surfaces, Hamiltonian stationary Lagrangian spheres 1 Introduction A Lagrangian submanifold M is an n-dimensional submanifold of a symplectic manifold (M 2n,ω) on which the symplectic form ω vanishes. If (M,ω) carries a Käehler structure, that is, it has an integrable almost complex structure J such that the bilinear form.,. = ω(.,j.) defines a Riemannian metric, the associated Riemannian properties of the Lagrangian submanifolds have been studied by many authors. From a Riemannian point of view, when we focus our attention on variational problems related with the volume functional, its critical points, i.e. the minimal Lagrangian submanifolds, play an important role (special Lagrangian submanifolds, etc. ). On the other hand, from a symplectic point of view, the intersection properties of Lagrangian submanifolds under Hamiltonian deformations have been studied in depth (Arnold s conjecture, etc. ). Motivated by these facts, Y.-G. Oh raised in [22] the problem of volume minimizations of Lagrangian submanifolds under Hamiltonian deformations in Kähler manifolds. More precisely, one only considers normal variations V along M such that the one form α V = JV,. T M is exact. In this way, the notion of Hamiltonian minimal (H-minimal for brevity) or Hamiltonian stationary Lagrangian submanifold was introduced in 1993 (see [22]) as the critical points of the volume functional for all Hamiltonian isotopy of the Lagrangian submanifold. The Euler-Lagrangian equation of this variational problem is δα H = 0, (1) where H is the mean curvature vector of the submanifold, α H is the Maslov form and δ is the Hodge-dual of the exterior derivative d on M with respect to the induced metric. In particular, minimal Lagrangian submanifolds (i.e. with vanishing mean curvature vector) and, more generally, Lagrangian submanifolds with parallel mean curvature vector are trivially H-minimal ones. Even when M is a simply-connected complex space form, not many examples of H-minimal Lagrangian submanifolds are known outside the class of Lagrangian submanifolds with parallel mean curvature vector. A chronological journey about the discovery of examples can be the following: In 1998 it was classified in [8] the S 1 -invariant H-minimal Lagrangian tori in complex Euclidean plane C 2. H-minimal Lagrangian cones in C 2 were studied in 1999, see [23]. In 2000 and 2002 (see [13] and [14]) a Weierstrass type representation formula was derived to describe all H-minimal Lagrangian tori and Klein bottles in C 2. When the ambient space is the complex projective plane CP 2 or the complex hyperbolic plane CH 2, conformal parametrizations of H-minimal Lagrangian surfaces using holomorphic data were obtained in 2002 in [15]. Making use of this technique, in 2003 H-minimal Lagrangian symply periodic cylinders

2 44 Castro and H-minimal Lagrangian surfaces with a non conical singularity in C 2 were constructed in [1]. Also in 2003 some equations for H-minimal Lagrangian surfaces in CP 2 were obtained in [20] and the particular solutions in the case of tori were given. In 2004 we found in [21] some examples of H-minimal Lagrangian product submanifolds of arbitrary dimension in C n and CP n. In 2005 it was proved in [16] and [18] that any H-minimal Lagrangian torus in CP 2 can be obtained by solving certain Hamiltonian ODEs on finite dimensional subspaces of a twisted loop algebra over su(3). In 2006 there were more papers devoted to the construction of examples: in [19] the authors gave a family of H-minimal Lagrangian tori in CP 2 with S 1 -symmetry and in [5] H-minimal Lagrangian surfaces in C 2 were constructed using a spherical curve and a hyperbolic curve such that their curvature functions (in terms of the arc parameter) are linear. Also in 2006 a classification of H-minimal Lagrangian submanifolds foliated by (n 1)-spheres in C n was given in [2] and, finally, the aim of [6] was the construction of H-minimal Lagrangian product submanifolds in complex Euclidean space C n, the complex projective space CP n and the complex hyperbolic space CH n, for arbitrary n 2. The examples in CP n were constructed by projections, via the Hopf fibration Π : S 2n+1 CP n, of certain family of Legendrian submanifolds of the sphere S 2n+1. The cones with links in this family of Legendrian submanifolds provided new examples of H-minimal Lagrangian submanifolds in C n+1. Using the Hopf fibration Π : H 2n+1 1 CH n and a similar family of Legendrian submanifolds of the anti De Sitter space H 2n+1 1, the authors also found out the examples of H-minimal Lagrangian submanifolds in CH n. We draw reader s attention to the fact that there are not any examples of non trivial Lagrangian H- minimal spheres in the above literature about this subject. To find the reason we recall the formula dα H = Φ S, (2) first proved by P. Dazord [10] for a Lagrangian immersion Φ : M M in a Kähler manifold, where S is the Ricci form of M. In particular, if M is Einstein Kähler, i.e. S = cω where c is a constant, then the Maslov form α H is closed. In this case, (1) is equivalent to α H = 0, where = dδ + δd. In this way, if the first Betti number of M is zero, then M must be minimal. Thus H-minimal Lagrangian spheres in Kähler Einstein manifolds (in particular, in complex space forms) are trivial, that is, necessarily minimal. It is well known that there do not exist compact minimal submanifolds neither in C n nor CH n. Examples of minimal Lagrangian spheres in Kähler Einstein 4-manifolds are the standard totally geodesic Lagrangian sphere in CP 2 or the totally geodesic Lagrangian sphere x (x, x) in S 2 S 2 (see [9]). In fact, they are the only minimal Lagrangian spheres in CP 2 and S 2 S 2 respectively. This article is a part of a joint paper with F. Torralbo and F. Urbano (see [7]) wherein we are interested in non trivial (i.e. non minimal) Hamiltonian minimal Lagrangian orientable compact surfaces with the easiest topology: spheres. Hence we must look for them in non-einstein Kähler surfaces. For the sake of simplicity, we will consider as ambient manifold the product of two Riemann surfaces. In [9] the authors deal with the minimal Lagrangian surfaces of the Einstein-Kähler surface S 2 S 2, studying local geometric properties and showing that they can be locally described as Gauss maps of minimal surfaces in S 3 R 4. They also discuss the second variation of the area and characterize the most relevant examples by their stability behaviour. As a first target 4-manifold we will take the product of two spheres too, but of different radii in order to get a non-einstein Kähler surface. As usual, S 2 denotes the unit sphere in R 3 and, along the paper, S 2 (r) = {(z,x) C R/ z 2 + x 2 = r 2 } will denote the sphere of radius r in R 3. 2 Main results Our main contribution is the following Existence and Uniqueness Theorem. Theorem 1 Let Φ : Σ S 2 (r 1 ) S 2 (r 2 ) be a Hamiltonian minimal Lagrangian immersion of a sphere Σ in the product of two spheres of radii r 1 < r 2.

3 Symposium Valenciennes 45 Then Φ(Σ) is congruent to the Hamiltonian minimal Lagrangian sphere S 0 = {((z 1,x 1 ),(z 2,x 2 )) S 2 (r 1 ) S 2 (r 2 ) (C R) 2 / (3) R(z 1 z 2 ) = 0, r 1 x 1 r 2 x 2 = r1 2 }, that can be regularly parametrized by S 0 = Φ 0 (S 2 ), with Φ 0 = (φ 0,ψ 0 ) : S 2 S 2 (r 1 ) S 2 (r 2 ) (C R) 2 φ 0 (x,y,z) = 2 c 1 c 2 c 1 c 2 z 2 ψ 0 (x,y,z) = 2 c 1 c 2 c 1 c 2 z 2 ( iz(x + iy), c ) 1 + (2c 1 c 2 )z 2 2 c 1 c1 c 2 ( x iy, (c ) 1 2c 2 ) + c 2 z 2 2 c 2 c1 c 2 (4) where c 1 = 1/r 2 1 and c 2 = 1/r 2 2. Figure 1: Images of the meridians and paralles of S 2 by (φ δ,ψ δ ) for δ = 0 and δ > 0 The sphere S 0 belongs to a one-parameter family of Lagrangian spheres in S 2 (r 1 ) S 2 (r 2 ) which are minimizers of a Willmore-type functional and that we collect in the following result. Theorem 2 Let Φ : Σ S 2 (r 1 ) S 2 (r 2 ) be a Lagrangian immersion of a sphere Σ in the product of two spheres of radii r 1 < r 2. Then Z ( H 2 + r2 1 + ) Σ 4r1 2 da 8π, and the equality holds if and only if Φ(Σ) is congruent to some Lagrangian sphere in the one-parameter

4 46 Castro family {S δ /δ 0}, with where s δ = sinhδ and c δ = coshδ. S δ = {((z 1,x 1 ),(z 2,x 2 )) S 2 (r 1 ) S 2 (r 2 ) (C R) 2 / R(z 1 z 2 ) = s δ c δ ( z 2 2 z 1 2 ), r 1 x 1 r 2 x 2 = r 2 1 }, (5) Moreover, the Lagrangian spheres S δ, δ 0, can be regularly parametrized by S δ = Φ δ (S 2 ), with Φ δ = (φ δ,ψ δ ) : S 2 S 2 (r 1 ) S 2 (r 2 ) (C R) 2 φ δ (x,y,z) = 2 c 1 c 2 (c 1 c 2 δ c 2s 2 δ ) + (c 1s 2 δ c 2c 2 δ )z2 ((s δ + ic δ z)(x + iy), (c 1c 2 δ 2c 1 c 2 s 2 δ ) + (c 1s 2 δ + 2c 1 c 2 c 2 ) δ )z2 2 c 1 c1 c 2 (6) where c 1 = 1/r 2 1 and c 2 = 1/r 2 2. ψ δ (x,y,z) = 2 c 1 c 2 (c 1 c 2 δ c 2s 2 δ ) + (c 1s 2 δ c 2c 2 δ )z2 ((c δ + is δ z)(x iy), (c 1c 2 δ 2c 2 c 2 s 2 δ ) + (c 1s 2 δ + 2c 2 c 2 c 2 ) δ )z2 2 c 2 c1 c 2 Among the geometric properties of the Lagrangian spheres {S δ /δ 0}, we emphasize the following: Φ δ, δ 0, is a Hamiltonian deformation of Φ 0 : the normal component of the variational vector field d dδ δ=0 Φ δ (S 2 ) is given by J h, for a certain smooth function h : S 2 R. Here denotes the gradient with respect to the induced metric. The H-minimal sphere S 0 is (Hamiltonian) unstable: precisely Q(h) < 0 where Q is the quadratic form associated to the second variation formula for H-minimal Lagrangian surfaces (see [22]). The area (with the induced metric) of the spheres S δ, δ 0, is a strictly decreasing function that goes asymptotically to 0 when δ. As a consequence, there do not exist area minimizers in the isotopy class of the Hamiltonian minimal Lagrangian sphere S 0 in S 2 (r 1 ) S 2 (r 2 ), r 1 r 2. 3 Outline of the proofs 3.1 Lagrangian surfaces in the product of two spheres We consider S 2 (r i ), i = 1,2, with its standard Euclidean metric, and its structure of Riemann surface given by J i xv = 1 r i x v, for any v T x S 2 (r i ), x S 2 (r i ), where stands for the vectorial product in R 3, i = 1,2. Its Kähler 2-form is the area 2-form ω i defined by ω i (v,w) = J i xv,w = 1 r i det{x,v,w} for any v,w T x S 2 (r i ), i = 1,2. We endow S 2 (r 1 ) S 2 (r 2 ) with the product metric (also denoted by, ) and the product complex structure J = (J 1,J 2 ) which becomes S 2 (r 1 ) S 2 (r 2 ) in a Kähler surface (Einstein Kähler if and only if r 1 = r 2 ). Its Kähler 2-form is ω = π 1 ω 1 + π 2 ω 2 where π i, i = 1,2, are the projections of S 2 (r 1 ) S 2 (r 2 ) onto S 2 (r i ), i = 1,2. Let Φ = (φ,ψ) : Σ S 2 (r 1 ) S 2 (r 2 ) be an immersion of a surface Σ and denote by g = φ, + ψ, the induced metric. The immersion Φ is said to be Lagrangian if Φ ω = 0, i.e. φ ω 1 + ψ ω 2 = 0. This means that 0 = JdΦ p (v),dφ p (w) = J 1 dφ p (v),dφ p (w) + J 2 dψ p (v),dψ p (w),

5 Symposium Valenciennes 47 for any p Σ and v,w T p Σ. If Φ = (φ,ψ) : Σ S 2 (r 1 ) S 2 (r 2 ) is an isometric immersion of an oriented surface with area 2-form ω Σ, we can define the Jacobians of φ and ψ by φ ω 1 = Jac(φ)ω Σ, ψ ω 2 = Jac(ψ)ω Σ. Hence, when Σ is oriented, Φ is Lagrangian if and only if Jac(φ) = Jac(ψ). We will call the function C := Jac(φ) = Jac(ψ) (7) the associated Jacobian of the oriented Lagrangian surface Σ. It is not difficult to get that C ranges between 0 and 1/4. The case C 0 characterize the simplest examples: the product of two (spherical) curves. From (2) we obtain that dα H = 1 2 ( 1 1 )C ω r1 2 Σ. Therefore if Φ has parallel mean curvature vector (or, in particular, Φ is minimal), then Φ must be a product of circles. These tori are trivial examples of H-minimal Lagrangian surfaces in S 2 (r 1 ) S 2 (r 2 ), r 1 r 2. The Gauss equation of Φ can be written as ( 1 K = r )C 2 r H 2 σ 2 2, (8) where K is the Gauss curvature of Σ, H the mean curvature of Φ and σ the second fundamental form of Φ. 3.2 Ideas of the proofs From now on, we work with a Hamiltonian minimal Lagrangian sphere Σ in the product of two spheres of radii r 1 < r 2, that is the hypothesis of Theorem 1. The key geometric tool for the proof of Theorem 1 is that we are able to associate to a H-minimal Lagrangian surface in S 2 (r 1 ) S 2 (r 2 ), r 1 r 2, a pair of quadratic Hopf differentials involving the mean curvature vector and the product structure of the ambient manifold (see [7]) following the same spirit that [3] and [4]. The fact that both differentials must be zero on a sphere allows us a good control of H and the associated Jacobian defined in (7), that in this context are related by H 2 = r2 1 4r C 2. (9) Then we check that the nonnegative function H on Σ is of absolute value type (see [11]), that is, there exists a function t of holomorphic type on Σ (i.e. locally t = t 0 t 1 where t 0 is holomorphic and t 1 smooth without zeroes) such that H = t. So the zeroes of H are isolated, and outside its zeroes, the function H is smooth. Using Lemma 4.1 in [11] we have that R Σ log H da = 2πN( H ), where N( H ) is the sum of all orders for all zeroes of H. After long computations we obtain ( ) 2 H r2 1 (r2 1 + ) H 2 (r1 log H = 2 )2 2 Using (8), (9) and (10) in the above integral formula, we arrive at Z ( H 2 + r2 1 + ) Σ 4r1 2 da = 2π(2 + N( H )). (11) Then JH is a tangent vector field to a Lagrangian sphere Σ S 2, whose zeroes are isolated. Using Poincaré-Hopf theory we have that N( H ) = χ(s 2 ) = 2. Finally we get from (11) that our Lagrangian H-minimal sphere must satisfy Z ( H 2 + r2 1 + ) Σ 4r1 2 da = 8π. (12) This finishes the first part of the proof of Theorem 1. (10)

6 48 Castro 3.3 Willmore spheres in R 6 Now we are going to prove Theorem 2 in order to conclude the proof of the main Theorem 1. Let Φ : Σ S 2 (r 1 ) S 2 (r 2 ) be a Lagrangian immersion of a sphere Σ in the product of two spheres of radii r 1 < r 2. We start by recalling a result of [24] that we need for this purpose. Let Φ : Σ R n be an immersion of a compact surface Σ with mean curvature vector H and maximum multiplicity µ, i.e. there exist µ points {p 1,..., p µ } on Σ such that Φ(p i ) = a, for all 1 i µ. Then Z Σ H 2 da 4πµ, and the equality holds if and only if H is given on Σ = Σ {p 1,..., p µ } by H = 2(Φ a), where Φ a 2 Φ a stands for normal component. This condition about the mean curvature H means that : Σ R n Φ a 2 is a minimal immersion. If we compute the mean curvature vector H of Φ = (φ,ψ) : Σ R 6, we get that ( φ 2 H = 2H r1 2, ψ ) r2 2 and so H 2 = H 2 + r Thus we have that the integral of (12) is nothing but the Willmore functional 4r1 2 of the immersion Φ looked at R 6. By topological reasons (see [7]), as Σ is a sphere Φ can not be an embedding and then µ 2. Thus applying the above result we get the inequality of Theorem 2. If the equality holds (this was our end point of the first part of the proof of Theorem 1) we arrive at that µ = 2, Φ(p 1 ) = Φ(p 2 ) = a = (a 1,a 2 ) S 5 (r), r 2 = r 2 1 +, and Σ must be the compactification of a complete minimal surface Σ \ {p 1, p 2 } in R 6 with two planar ends and total curvature R Σ ( H 2 K)dA = 4π. We write Φ = Φ a Φ a 2 : Σ \ {p 1, p 2 } R 6. Using that Φ a 2 = 2(r 2 φ,a ) and (13) we obtain that Φ,â = r 2 2 r2 1, where â = ( a 1,a 2 ). Now it is not difficult to get that Hence Φ lies in a 4-dimensional affine subspace of R 6. (13) Φ,a = 1 2, Φ,â = 0. (14) But then these examples are described as the family of complex surfaces of C 2 with finite total curvature 4π given in Proposition 6.6 in [17]. By choosing a i = (0,0,r i ), i = 1,2, we can prove that ( Φ = F, 1,G, 1 ) 4r 1 4r 2 where, up to congruences and dilations, (F,G) : Σ \ {p 1, p 2 } C 2 must belong to the one-parameter family of Lawlor s examples ([12]) given by F 2 G 2 = 1, R(FG) = sinh2δ, δ 0. 2 From [5] we make use of the representation (F,G) : R S 1 C 2 given by (F,G)(t,s) = r2 1 ( (sinhδcosht + i coshδsinht)e is,(coshδcosht + i sinhδsinht)e is). (16) r 1 r 2 Now we can recover Φ from (14), (15) and (16). After a straightforward computation, using the map R S 1 S 2, (x + iy,z) = ( eis cosht,tanht) we arrive at the one-parameter family of examples Φ δ given in (15)

7 Symposium Valenciennes 49 (6). It is an exercise to check that all of them are Lagrangian immersions of spheres, whose intrinsic equations are given in (5). The proof of Theorem 1 finishes when we compute the divergence of the tangent vector field JH for each sphere S δ, obtaining (divjh)(x,y,z) = (r2 1 ) sinh2δz 2r1 2 (1 +, z2 ) that proves that the immersion Φ 0 given in (4) is the only H-minimal one in the family. 4 Directions of research We think that it is also interesting the study of Hamiltonian minimal Lagrangian spheres not only in the product of spheres of different radii but also in the product of two orientable Riemannian surfaces Σ i, i = 1,2, of constant curvature c i, i = 1,2, with c 1 > c 2. This is precisely the aim of [7]. References [1] H. Anciaux. Construction of many Hamiltonian stationary Lagrangian surfaces in Euclidean fourspace. Calc. of Var , [2] H. Anciaux, I. Castro and P. Romon. Lagrangian submanifolds foliated by (n 1)-spheres in R 2n. Acta Math. Sinica , [3] U. Abresch and H. Rosenberg. A Hopf differential for constant mean curvature surfaces in S 2 R and H 2 R. Acta Math , [4] U. Abresch and H. Rosenberg. Generalized Hopf differentials. Mat. Contemp , [5] I. Castro and B.-Y. Chen. Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves. Tohoku Math. J , [6] I. Castro, H. Li and F. Urbano. Hamiltonian-minimal Lagrangian submanifolds in complex space forms. Pacific Journal Math , [7] I. Castro, F. Torralbo and F. Urbano. Hamiltonian stationary Lagrangian spheres in non-einstein Kähler surfaces. Preprint [8] I. Castro and F. Urbano. Examples of unstable Hamiltonian-minimal Lagrangian tori in C 2. Compositio Math , [9] I. Castro and F. Urbano. Minimal Lagrangian surfaces in S 2 S 2. To appear in Comm. Anal. Geom. 15, [10] P. Dazord. Sur la geometrie des sous-fibres et des feuilletages lagrangiense. Ann. Sci. Ec. Norm. Super. IV, Ser , [11] J.-H. Eschenburg, I.V. Guadalupe and R.A. Tribuzy. The fundamental equations of minimal surfaces in CP 2. Math. Ann (1985). [12] G. Lawlor. The angle criterion. Invent. Math , [13] F. Hélein and P. Romon. Weierstarss representation of Lagrangian surfaces in four-dimensional space using spinors and quaternions. Comm. Math. Helvetici , [14] F. Hélein and P. Romon. Hamiltonian stationary Lagrangian surfaces in C 2. Comm. Anal. Geom , [15] F. Hélein and P. Romon. Hamiltonian stationary Lagrangian surfaces in Hermitian symmetric spaces. Differential Geometry and Integrable Systems (Tokyo, 2000), , Contemp. Math. 308, Amer. Math. Soc., Providence, RI, 2002.

8 50 Castro [16] F. Hélein and P. Romon. Hamiltonian stationary tori in complex projective plane. Proc. London Math. Soc , [17] D.A. Hoffman and R. Osserman. The geometry of the generalized Gauss map. Mem. Amer. Math. Soc. 236, [18] H. Ma. Hamiltonian stationary Lagrangian surfaces in CP 2. Ann. Global Anal. Geom , [19] H. Ma and M. Schmies. Examples of Hamiltonian stationary Lagrangian tori in CP 2. Geom. Dedicata , [20] A. E. Mironov. On Hamiltonian-minimal Lagrangian tori in CP 2. Sib. Math. J. 44,2003. [21] A. E. Mironov. On new examples of Hamiltonian-minimal and minimal Lagrangian submanifolds in C n and CP n. (Russian) Mat. Sb , [22] Y.G. Oh. Volume minimization of Lagrangian submanifolds under Hamiltonian deformations. Math. Z , [23] R. Schoen and J. Wolfson, Minimizing volume among Lagrangian submanifolds. Proc. Symp. Pure Math. 65, [24] L. Simon. Existence of surfaces minimizing the Willmore functional. Comm. Anal. Geom , 1993.