Lagrangian submanifolds in complex space forms satisfying an improved equality involving δ(2,2)

arXiv:1307.3968v1 [math.DG] 15 Jul 2013 LAGRANGIAN SUBMANIFOLDS IN COMPLEX SPACE FORMS SATISFYING AN IMPROVED EQUALITY INVOLVING δ(2,2) BANG-YEN CHEN, ALICA PRIETO-MAR´ IN, AND XIANFENG WANG Abstract. It was proved in [8, 9] that every Lagrangian submanifold Mof a complex space form ˜ M5(4c) of constant holomorphic sectional curvature 4csatisfies the following optimal inequality: δ(2,2) ≤25 4H2+ 8c, (A) where H2is the squared mean curvature and δ(2,2) is a δ-invariant on Mintroduced by the first author. This optimal inequality improves a special case of an earlier inequality obtained in [B.-Y. Chen, Japan. J. Math. 26 (2000), 105–127]. The main purpose of this paper is to classify Lagrangian submanifolds of ˜ M5(4c) satisfying the equality case of the improved inequality (A). 1. Introduction Let ˜ Mnbe a K¨ahler n-manifold with the complex structure J, a K¨ahler metric gand the K¨ahler 2-form ω. An isometric immersion ψ:M→˜ Mnof a Riemannian n-manifold Minto ˜ Mnis called Lagrangian if ψ∗ω= 0. Let ˜ Mn(4c) denote a K¨ahler n-manifold with constant holomorphic sectional curvature 4c, called a complex space form. A complete simply-connected complex space form ˜ Mn(4c) is holomorphically isometric to the complex Euclidean n-plane Cn, the complex projective n-space CPn(4c), or a complex hyperbolic n-space CHn(4c) according to c= 0, c > 0 or c < 0, respectively. B.-Y. Chen introduced in 1990s new Riemannian invariants δ(n1,...,nk). For any n-dimensional submanifold Min a real space form Rm(c) of constant 1991 Mathematics Subject Classification. Primary: 53C40; Secondary 53D12. Key words and phrases. Lagrangian submanifold, improved inequality, δ-invariants, ideal submanifolds, H-umbilical Lagrangian submanifold. A portion of this work was done while the second author was visiting Michigan State University in 2011, supported by a Fundaci´on C´amara grant, University of Sevilla, Spain. The third author was supported by the NSFC No.11171175 and the “Fundamental Research Funds for the Central Universities”. 1 2 B.-Y. CHEN, A. PRIETO-MART´ IN, AND X. WANG curvature c, he proved the following sharp general inequality (see [5, 7] for details): δ(n1,...,nk)≤n2(n+k−1−Pnj) 2(n+k−Pnj)H2 +1 2n(n−1) − k X j=1 nj(nj−1)c. (1.1) For Lagrangian submanifolds in a complex space form ˜ Mn(4c), we have Theorem A. Let Mbe an n-dimensional Lagrangian submanifold in a complex space form ˜ Mn(4c)of constant holomorphic sectional curvature 4c. Then inequality (1.1) holds for each k-tuple (n1,...,nk)∈ S(n). The following result from [6] extends a result in [10] on δ(2). Theorem B. Every Lagrangian submanifold of a complex space form ˜ Mn(4c)is minimal if it satisfies the equality case of (1.1) identically. Theorem B was improved recently in [8, 9] to the following inequality. Theorem C. Let Mbe an n-dimensional Lagrangian submanifold of ˜ Mn(4c). Then, for an (n1,...,nk)∈ S(n)with Pk i=1 ni< n, we have δ(n1,...,nk)≤n2nn−Pk i=1ni+ 3k−1−6Pk i=1(2 + ni)−1o 2nn−Pk i=1ni+ 3k+ 2−6Pk i=1(2 + ni)−1oH2 +1 2nn(n−1) −Xk i=1ni(ni−1)oc. (1.2) The equality sign holds at a point p∈Mif and only if there is an orthonormal basis {e1,...,en}at psuch that the second fundamental form h satisfies h(eαi, eβi) = Xγi hγi αiβiJeγi+3δαiβi 2 + ni λJeN+1,Xni αi=1hγi αiαi= 0, h(eαi, eαj) = 0, i 6=j;h(eαi, eN+1) = 3λ 2 + ni Jeαi, h(eαi, eu) = 0, h(eN+1, eN+1) = 3λJeN+1, h(eN+1, eu) = λJeu, N =n1+···+nk, h(eu, ev) = λδuvJeN+1, i, j = 1,...,k;u, v =N+ 2,...,n. (1.3) For simplicity, we call a Lagrangian submanifold of a complex space form δ(n1,...,nk)-ideal (resp., improved δ(n1,...,nk)-ideal) if it satisfies the equality case of (1.1) (resp., the equality case of (1.2)) identically. For k= 2 and n1=n2= 2, Theorem C reduces to the following. LAGRANGIAN SUBMANIFOLDS 3 Theorem D. Let Mbe a Lagrangian submanifold in a complex space form ˜ M5(4c)of constant holomorphic sectional curvature 4c. Then we have δ(2,2) ≤25 4H2+ 8c. (1.4) If the equality sign of (1.4) holds identically, then with respect some suitable orthonormal frame {e1,...,e5}the second fundamental form hsatisfies h(e1, e1) = αJe1+βJe2+µJe5, h(e1, e2) = βJe1−αJe2, h(e2, e2) = −αJe1−βJe2+µJe5, h(e3, e3) = γJe3+δJe4+µJe5, h(e3, e4) = δJe3−γJe4, h(e4, e4) = −γJe3−δJe4+µJe5, h(e5, e5) = 4µJe5, h(ei, e5) = µJei, i ∈∆; h(ei, ej) = 0, otherwise, (1.5) for some functions α, β, γ, δ, µ, where ∆ = {1,2,3,4}. The classification of δ(2,2)-ideal Lagrangian submanifolds in complex space forms ˜ M5(4c) is done in [13]. In this paper we classify improved δ(2,2)-ideal Lagrangian submanifolds in ˜ M5(4c). The main results of this paper are stated as Theorem 6.1, Theorem 7.1 and Theorem 8.1. 2. Preliminaries 2.1. Basic formulas. Let ˜ Mn(4c) denote a complete simply-connected K¨ahler n-manifold with constant holomorphic sectional curvature 4c. Then ˜ Mn(4c) is holomorphically isometric to the complex Euclidean n-plane Cn, the complex projective n-space CP n(4c), or a complex hyperbolic n-space CHn(−4c) according to c= 0, c > 0 or c < 0. Let Mbe a Lagrangian submanifold of ˜ Mn(4c). We denote the Levi- Civita connections of Mand ˜ Mn(4c) by ∇and ˜ ∇, respectively. The formulas of Gauss and Weingarten are given respectively by (cf. [7]) ˜ ∇XY=∇XY+h(X, Y ),˜ ∇Xξ=−AξX+DXξ, (2.1) for tangent vector fields Xand Yand normal vector fields ξ, where his the second fundamental form, Ais the shape operator and Dis the normal connection. The second fundamental form and the shape operator are related by hh(X, Y ), ξi=hAξX, Y i. 4 B.-Y. CHEN, A. PRIETO-MART´ IN, AND X. WANG The mean curvature vector −→ Hof Mis defined by −→ H=1 ntrace hand the squared mean curvature is given by H2=h−→ H, −→ Hi. For Lagrangian submanifolds, we have (cf. [7, 12]) DXJY =J∇XY, (2.2) AJX Y=−Jh(X, Y ) = AJY X. (2.3) Formula (2.3) implies that hh(X, Y ), JZiis totally symmetric. The equations of Gauss and Codazzi are given respectively by hR(X, Y )Z, Wi=Ah(Y,Z)X, W −Ah(X,Z)Y, W (2.4) +c(hX, W ihY, Zi−hX, ZihY, W i), (∇Xh)(Y, Z) = (∇Yh)(X, Z),(2.5) where Ris the curvature tensor of Mand ∇his defined by (∇Xh)(Y, Z) = DXh(Y, Z)−h(∇XY, Z)−h(Y, ∇XZ).(2.6) For an orthonormal basis {e1,...,en}of TpM, we put hi jk =hh(ej, ek), Jeii, i, j, k = 1,...,n. It follows from (2.3) that hi jk =hj ik =hk ij. 2.2. δ-invariants. Let Mbe a Riemannian n-manifold. Denote by K(π) the sectional curvature of a plane section π⊂TpM,p∈M. For any orthonormal basis e1,...,enof TpM, the scalar curvature τat pis τ(p) = Pi<j K(ei∧ej). Let Lbe a r-subspace of TpMwith r≥2 and {e1,...,er}an orthonormal basis of L. The scalar curvature τ(L) of Lis defined by τ(L) = X α<β K(eα∧eβ),1≤α, β ≤r. (2.7) For given integers n≥3, k≥1, we denote by S(n, k) the finite set consisting of k-tuples (n1,...,nk) of integers satisfying 2 ≤n1,···, nk< n and Pk j=1 i≤n. Put S(n) = ∪k≥1S(n, k). For each k-tuple (n1,...,nk)∈ S(n), the first author introduced in 1990s the Riemannian invariant δ(n1,...,nk) by δ(n1,...,nk)(p) = τ(p)−inf{τ(L1) + ···+τ(Lk)}, p ∈M, (2.8) where L1,...,Lkrun over all kmutually orthogonal subspaces of TpMsuch that dim Lj=nj, j = 1,...,k (cf. [7] for details). LAGRANGIAN SUBMANIFOLDS 5 2.3. Horizontal lift of Lagrangian submanifolds. The following link between Legendrian submanifolds and Lagrangian submanifolds is due to [16] (see also [7, pp. 247–248]). Case (i): CPn(4). Consider Hopf’s fibration π:S2n+1 →CPn(4).For a given point u∈S2n+1(1), the horizontal space at uis the orthogonal complement of ıu, ı = √−1,with respect to the metric on S2n+1 induced from the metric on Cn+1. Let ι:N→CPn(4) be a Lagrangian isometric immersion. Then there is a covering map τ:ˆ N→Nand a horizontal immersion ˆι:ˆ N→S2n+1 such that ι◦τ=π◦ˆι. Thus each Lagrangian immersion can be lifted locally (or globally if Nis simply-connected) to a Legendrian immersion of the same Riemannian manifold. In particular, a minimal Lagrangian submanifold of CPn(4) is lifted to a minimal Legendrian submanifold of the Sasakian S2n+1(1). Conversely, suppose that f:ˆ N→S2n+1 is a Legendrian isometric immersion. Then ι=π◦f:N→CPn(4) is again a Lagrangian isometric immersion. Under this correspondence the second fundamental forms hf and hιof fand ιsatisfy π∗hf=hι. Moreover, hfis horizontal with respect to π. Case (ii): CHn(−4). We consider the complex number space Cn+1 1equipped with the pseudo-Euclidean metric: g0=−dz1d¯z1+Pn+1 j=2 dzjd¯zj. Consider H2n+1 1(−1) = {z∈C2n+1 1:hz, zi=−1}with the canonical Sasakian structure, where h,iis the induced inner product. Put T′ z={u∈Cn+1 :hu, zi= 0}, H1 1={λ∈C:λ¯ λ= 1}.Then there is an H1 1-action on H2n+1 1(−1), z7→ λz and at each point z∈H2n+1 1(−1), the vector ξ=−ızis tangent to the flow of the action. Since the metric g0is Hermitian, we have hξ, ξi=−1. The quotient space H2n+1 1(−1)/∼, under the identification induced from the action, is the complex hyperbolic space CHn(−4) with constant holomorphic sectional curvature −4 whose complex structure Jis induced from the complex structure Jon Cn+1 1via Hopf’s fibration π:H2n+1 1(−1) →CHn(4c). Just like case (i), suppose that ι:N→CHn(−4) is a Lagrangian immersion, then there is an isometric covering map τ:ˆ N→Nand a Legendrian immersion f:ˆ N→H2n+1 1(−1) such that ι◦τ=π◦f. Thus every Lagrangian immersion into CHn(−4) an be lifted locally (or globally if Nis 6 B.-Y. CHEN, A. PRIETO-MART´ IN, AND X. WANG simply-connected) to a Legendrian immersion into H2n+1 1(−1). In particular, Lagrangian minimal submanifolds of CHn(−4) are lifted to Legendrian minimal submanifolds of H2n+1 1(−1). Conversely, if f:ˆ N→H2n+1 1(−1) is a Legendrian immersion, then ι=π◦f:N→CHn(−4) is a Lagrangian immersion. Under this correspondence the second fundamental forms hf and hιare related by π∗hf=hι. Also, hfis horizontal with respect to π. Let hbe the second fundamental form of Min S2n+1(1) (or in H2n+1 1(−1)). Since S2n+1(1) and H2n+1 1(−1) are totally umbilical with one as its mean curvature in Cn+1 and in Cn+1 1, respectively, we have ˆ ∇XY=∇XY+h(X, Y )−εL, (2.9) where ε= 1 if the ambient space is Cn+1; and ε=−1 if it is Cn+1 1. 3. H-umbilical Lagrangian submanifolds and complex extensors 3.1. H-umbilical Lagrangian submanifolds. Definition 3.1. A non-totally geodesic Lagrangian submanifold of a K¨ahler n-manifold is called H-umbilical if its second fundamental form satisfies h(ej, ej) = µJen, h(ej, en) = µJej, j = 1,...,n−1, h(en, en) = ϕJen, h(ej, ek) = 0,1≤j6=k≤n−1,(3.1) for some functions µ, ϕ with respect to an orthonormal frame {e1,...,en}. If the ratio of ϕ:µis a constant r, the H-umbilical submanifold is said to be of ratio r. If G:Nn−1→Enis a hypersurface of a Euclidean n-space Enand γ:I→C∗is a unit speed curve in C∗=C− {0}, then we may extend G:Nn−1→Ento an immersion I×Nn−1→Cnby γ⊗G:I×Nn−1→ C⊗En=Cn, where (γ⊗G)(s, p) = F(s)⊗G(p) for s∈I, p ∈Nn−1.This extension of Gvia tensor product ⊗is called the complex extensor of Gvia the generating curve γ. H-umbilical Lagrangian submanifolds in complex space forms were classified in a series of papers by the first author (cf. [2, 3, 4]). In particular, the following two results were proved in [2]. Theorem E. Let ι:Sn−1⊂Enbe the unit hypersphere in Encentered at the origin. Then every complex extensor of ιvia a unit speed curve LAGRANGIAN SUBMANIFOLDS 7 γ:I→C∗is an H-umbilical Lagrangian submanifold of Cnunless γis contained in a line through the origin (which gives a totally geodesic Lagrangian submanifold). Theorem F. Let Mbe an H-umbilical Lagrangian submanifold of Cn with n≥3. Then Mis either a flat space or congruent to an open part of a complex extensor of ι:Sn−1⊂Envia a curve γ:I→C∗. 3.2. Legendre curves. A unit speed curve z:I→S3(1) ⊂C2(resp., z:I→H3 1(−1) ⊂C2 1) is called Legendre if hz′,izi= 0. It was proved in [3] that a unit speed curve zin S3(1) (resp., in H3 1(−1)) is Legendre if and only if it satisfies z′′ = iλz′−z(resp., z′′ = iλz′+z) (3.2) for a real-valued function λ. It is known in [3] that λis the curvature function of zin S3(1) (resp., in H3 1(−1)) (see also [1, Lemmas 3.1 and 3.2]). 3.3. H-umbilical submanifolds with arbitrary ratio. We provide a general method to construct H-umbilical Lagrangian submanifolds with any given ratio in CPn(4) via curves in S2(1 2) (resp., in CHn(−4) via curves in H2(−1 2)). Proposition 3.2. For any real number rthere exist H-umbilical Lagrangian submanifolds of ratio rin CPn(4) and in CHn(−4). Proof. If r= 2 this was done in [3, Theorems 5.1 and 6.1]. If r6= 2, H- umbilical Lagrangian submanifolds of ratio rcan be constructed as follows: Case (a): CP n(4). Let S2(1 2) = {x∈E3;hx,xi=1 4}. The Hopf fibration πfrom S3(1) onto S2(1 2)≡CP1(4) is given by (cf. [1]) π(z1, z2) = z1¯z2,1 2(|z1|2−|z2|2),(z1, z2)∈S3(1) ⊂C2.(3.3) For a Legendre curve zin S3(1), the projection γz=π◦zis a curve in S2(1 2). Conversely, each curve γin S2(1 2) gives rise to a horizontal lift ˜γin S3(1) via πwhich is unique up to a factor eiθ, θ ∈R. Notice that each horizontal lift of γis a Legendre curve in S3(1). Moreover, since the Hopf fibration is a Riemannian submersion, each unit speed Legendre curve zin S3(1) is projected to a unit speed curve γzin S2(1 2) with the same curvature. 8 B.-Y. CHEN, A. PRIETO-MART´ IN, AND X. WANG It was known in [3, Lemma 7.2] that, for a given H-umbilical Lagrangian submanifold of ratio r6= 2 in ˜ Mn(4c), the function µin (3.1) satisfies µµ′′ −r−3 r−2µ′2+ (r−2)µ2((r−1)µ2+c) = 0.(3.4) If µis a non-trivial solution of (3.4) with c= 1, then there is a unit speed curve γin S2(1 2) whose curvature equals to rµ. Let zbe a horizontal lift of γin S3(1). Then zis a unit speed Legendre curve satisfying z′′(x) = irµz′(x)−z(x) (cf. [3, Theorem 4.1] or [1, Lemma 3.1]). Consider the map ψ:M5→S11(1) ⊂C6defined by ψ(x, y1,...,y5) = (z1(x), z2(x)y1,...,...,z2(x)y5), 5 X j=1 y2 j= 1.(3.5) It follows from [3, Theorem 4.1 and Lemma 7.2] that π◦ψis a H-umbilical Lagrangian submanifold of ratio rin CP n(4) such that h(ej, ej) = µJe5, h(ej, en) = µJej, h(en, en) = rµJen, h(ej, ek) = 0,1≤j6=k≤n−1,(3.6) with respect to suitable orthonormal frame {e1,...,e5}. Case (b): CHn(−4). For a non-trivial solution of (3.4) with c=−1, we can construct an H-umbilical Lagrangian submanifold of CHn(−4) via the Hopf fibration π:H3 1(−1) →CH1(−4) ≡H2(−1 2) in a similar way as case (a), where π(z1, z2) = z1¯z2,1 2(|z1|2+|z2|2),(z1, z2)∈H3 1(−1) ⊂C2 1,(3.7) and H2(−1 2) = {(x1, x2, x3)∈E3 1:x2 1−x2 2−x2 3=1 4, x1≥1 2}is the model of the real projective plane of curvature −4.  3.4. Classification of H-umbilical submanifolds of ratio 4. The equation of Gauss and (3.1) imply that H-umbilical Lagrangian submanifolds of ratio r6= 4 in complex space forms contain no open subsets of constant sectional curvature. Hence we conclude from [3, Theorems 4.1 and 7.1] and §3.3 the following results. Lemma 3.3. An H-umbilical Lagrangian submanifold Mof ratio 4 in CP5(4) is congruent to an open portion of π◦ψ, where π:S11(1) →CP 5(4) is Hopf’s fibration, ψ:M→S11(1) ⊂C6is given by ψ(t, y1,...,y5) = (z1(t), z2(t)y),{y∈E5:hy,yi= 1},(3.8) LAGRANGIAN SUBMANIFOLDS 9 and z= (z1, z2) : I→S3(1) ⊂C2is a unit speed Legendre curve satisfying z′′ = 4iµz′−z, and µis a nonzero solution of 2µµ′′ −µ′2+4µ2(3µ2+1) = 0. Let Mbe an H-umbilical Lagrangian submanifold in CH5(−4) satisfying (3.1). We may assume that µis defined on an open interval I∋0. Since H-umbilical submanifolds of ratio 4 in CH5(−4) contain no open subsets of constant curvature, Theorems 4.2 and 9.1 of [3] and results in §3.3 imply the following classification of H-umbilical submanifolds of ratio 4 in CH5(−4). Lemma 3.4. An H-umbilical Lagrangian submanifold Mof ratio 4 in CH5(−4) is congruent to an open part of π◦ψ, where π:H11 1(−1) → CH5(−4) is Hopf’s fibration and ψ:M→H11 1(−1) ⊂C6 1is either one of ψ(t, y1,...,y4) = (z1(t), z2(t)y),{y∈E5:hy,yi= 1},(3.9) ψ(t, y1,...,y4) = (z1(t)y, z2(t)),{y∈E5 1:hy,yi=−1},(3.10) where zis a unit speed Legendre curve in H3 1(−1) satisfying z′′ = 4iµz′+z and µis a non-trivial solution of 2µµ′′ −µ′2+ 4µ2(3µ2−1) = 0; or ψis ψ(t, u1,...,u4) = √µeiRt 0µ(t)dt 1 + 1 2 4 X j=1 u2 j−it+1 2µ−1 2µ(0), iµ(0) −µ′(0) 2µ(0) 1 2 4 X j=1 u2 j−it+1 2µ−1 2µ(0)!, u1,...,u4!, (3.11) where z= (z1, z2) : I→H3 1(−1) ⊂C2 1is a unit speed Legendre curve and µ is a non-trivial solution of µ′2= 4µ2(1 −µ2). Example. It is easy to verify that µ= sech 2tis a non-trivial solution of µ′2= 4µ2(1 −µ2). Using µ= sech 2t, (3.11) reduces to ψ(t, u1,...,u4) = ei tan−1(tanh t) √cosh 2t 1 2−it+1 2 4 X j=1 u2 j+cosh 2t 2, t−i 2+i 2 4 X j=1 u2 j+i cosh 2t 2, u1,...,u4!. (3.12) It is direct to verify that (3.12) satisfies hψ, ψi=−1 and the composition π◦ψgives rise to an H-umbilical Lagrangian submanifold of ratio 4 in CH5(−4). 16 B.-Y. CHEN, A. PRIETO-MART´ IN, AND X. WANG Since Mis a Lagrangian submanifold in C5, (6.4) and (6.6) show that iφis perpendicular to each tangent space of M. Hence φis a horizontal immersion in the unit hypersphere S9(1) ⊂C5. Moreover, it follows from (6.9) that the second fundamental form of φis the original second fundamental form of M respect to to the second factor N4of the warped product I×ρ(t)N4. Hence, φis a minimal horizontal immersion in S9(1). Therefore, φis a horizontal lift of a minimal Lagrangian immersion in CP 4(4). Now, it follows from (6.2) that φis a horizontal lift of a δ(2)-ideal minimal Lagrangian submanifold of CP4(4). By direct computation we find ˜ ∇eαL−e5 ν+ iµ= 0, α = 1,...,5.(6.10) Thus, by (6.4), up to translations the Lagrangian immersion Lis L=e−iϕ ν+ iµφ(u1,...,u4),(6.11) where φis a horizontal minimal immersion in S9(1) and ν, ϕ, µ satisfy dν dt =−3µ2−ν2,dϕ dt =−4µ, dµ dt = 2µν. (6.12) From (6.12) we find dν dµ +ν 2µ=−3µ 2ν.(6.13) After solving (6.13) we get ν=±pc2µ−1−µ2for some real number c > 0. Replacing e5by −e5if necessary, we have ν=pc2µ−1−µ2.(6.14) It follows from (6.12) an (6.14) that ϕ′(µ) = −2/pc2µ−1−µ2. By solving the last equation we find ϕ=−4 3i tan−1pµ3/(c2−µ3)+c0for some constant c0. Therefore, we have the theorem after applying a suitable translation in µ. Remark 6.2.Minimal δ(2,2)-ideal Lagrangian submanifolds in complex space forms C5,CP5and CH5are classified in [13]. Also δ(2)-ideal minimal Lagrangian submanifolds in CP 4and CH4have been classified recently in [14]. LAGRANGIAN SUBMANIFOLDS 17 Let γ(t) be a unit speed curve in C∗. We put γ(t) = r(t)eiθ(t), γ′(t) = eiζ(t).(6.15) The following result gives H-umbilical submanifolds of C5with ratio 4. Proposition 6.3. If Mis an H-umbilical Lagrangian submanifold of C5of ratio 4, then Mis an open part of a complex extensor γ⊗ιof the unit hypersphere ι:S4(1) ⊂E5via a generating curve γ:I→C∗whose curvature satisfies κ= 4θ′. Proof. If Mis an H-umbilical Lagrangian submanifold of C5with ratio 4, then the second fundamental form satisfies h(ej, ej) = µJe5, h(ej, e5) = µJej, j ∈∆, h(e5, e5) = 4µJe5, h(ej, ek) = 0,1≤j6=k≤4, for a nonzero function µ. Thus Gauss’ equation yields K(e1∧e5) = 3µ2. Hence Mis non-flat. Therefore, according to Theorem F, Mis an open part of a complex extensor of ι:Sn−1(1) ⊂Envia a generating curve γ:I→C∗. It follows from [2] that the functions ϕand µin (4.1) are related with the two angle functions ζand θby ϕ=ζ′(t) = κand µ=θ′(t). Thus whenever γis a unit speed curve satisfying κ= 4θ′, the complex extensor γ⊗ιis an H-umbilical Lagrangian submanifold of ratio 4. Conversely, every H- umbilical Lagrangian submanifold of ratio 4 in Cncan be obtained in such way.  7. Improved δ(2,2)-ideal Lagrangian submanifolds of CP5 Theorem 7.1. Let Mbe an improved δ(2,2)-ideal Lagrangian submanifold in CP5(4). Then it is one of the following Lagrangian submanifolds: (1) aδ(2,2)-ideal Lagrangian minimal submanifold; (2) an H-umbilical Lagrangian submanifold of ratio 4; (3) a Lagrangian submanifold defined by L(µ, u2,...,u4) = 1 c√µeiθφ, e3iθ(pc2−µ3−µ−iµ3 2),(7.1) where cis a positive real number, φ:N4→S9(1) ⊂C5is a horizontal lift of a non-totally geodesic δ(2)-ideal Lagrangian minimal immersion in CP4(4), and θ(µ)satisfies dθ dµ =1 2pc2µ−1−µ2−1.(7.2) 18 B.-Y. CHEN, A. PRIETO-MART´ IN, AND X. WANG Proof. Under the hypothesis there is an orthonormal frame {e1,...,e5}such that (4.1) holds. If µ= 0, then Mis a δ(2,2)-ideal Lagrangian minimal submanifold. Thus we obtain case (1). If µ6= 0 and a, b = 0, then Mis an H-umbilical Lagrangian submanifold of ratio 4, which gives case (2). Next, assume that a, µ 6= 0. Then Lemma 4.5 implies b= 0. So, by Lemmas 5.1 we obtain (5.1) and (5.2). Also, in this case Mis locally a warped product I×ρ(t)N4with e5=∂taccording to Lemma 5.3. From Lemma 4.1, we find h(e1, e1) = aJe1+µJe5, h(e1, e2) = −aJe2, h(e2, e2) = −aJe1+µJe5, h(e3, e3) = h(e4, e4) = µJe5, h(e5, e5) = 4µJe5, h(ei, e5) = µJei, i ∈∆, h(ei, ej) = 0, otherwise. (7.3) By Lemma 5.4 we have the following ODE system: dν dt =−1−ν2−3µ2,dµ dt = 2µν. (7.4) Let θ(t) be a function on Msatisfying θ′(t) = µ. (7.5) Let Ldenote the horizontal lift in S11(1) ⊂C6of the Lagrangian immersion of Min CP5(4) via Hopf ’s fibration. Consider the maps: ξ=e−3iθ(e5−(ν+ iµ)L) p1 + µ2+ν2, φ =e−iθ(L+ (ν−iµ)e5) p1 + µ2+ν2.(7.6) Then hξ, ξi=hφ, φi= 1. From ∇eje5=νej,j∈∆,and (7.4), we find ˜ ∇ejξ= 0. Moreover, it follows from Lemma 5.1 and (7.3) that ˜ ∇e5e5= 4iµe5−L. Thus we also hhve ˜ ∇e5ξ= 0. Hence ξis a constant unit vector in C6. Similarly, we also have ˜ ∇e5φ= 0. So φis independent of t. Therefore, by combining (7.6) we find L=eiθφ−e3iθ(ν−iµ)ξ p1 + µ2+ν2.(7.7) Since φis orthogonal to ξ, iξ, after choosing ξ= (0,...,0,1) ∈C6we obtain L=1 p1 + µ2+ν2eiθφ, e3iθ(ν−iµ)(7.8) LAGRANGIAN SUBMANIFOLDS 19 It follows from (7.4) and (7.5) that dν dµ =−1 + ν2+ 3µ2 2µν ,dθ dµ =1 2ν.(7.9) Solving the first differential equation in (7.9) gives ν=±pc2µ−1−µ2−1, c ∈R+.(7.10) By replacing e5by −e5if necessary, we have ν=pc2µ−1−µ2−1. Consequently, L=1 c√µeiθφ, e3iθ(pc2−µ3−µ−iµ3 2),(7.11) It follows from (5.1), (7.3) and the second formula in (7.6) that ˆ ∇ejφ=ce−iθ √µej, j ∈∆.(7.12) Thus after applying (6.11) and (7.12) we derive that ˆ ∇eβˆ ∇eαφ= n X γ=2 Γk ij + ihk ijφ∗(ek)−hφ∗(ei), φ∗(ej)iφ, i, j ∈∆.(7.13) Hence φis a horizontal immersion in S9(1). Moreover, it follows from (7.13) that the second fundamental form of φis a scalar multiple of the original second fundamental form of Mrestricted to the second factor of the warped product I×ρN. Consequently, φis a minimal horizontal immersion in S9(1) of a non-totally geodesic δ(2)-ideal Lagrangian minimal submanifold of CP4(4). The converse is easy to verify.  8. Improved δ(2,2)-ideal Lagrangian submanifolds of CH5 Theorem 8.1. Let Mbe an improved δ(2,2)-ideal Lagrangian submanifold in CH5(−4). Then Mis one of the following Lagrangian submanifolds: (i) aδ(2,2)-ideal Lagrangian minimal submanifold; (ii) an H-umbilical Lagrangian submanifold of ratio 4; (iii) a Lagrangian submanifold defined by L(µ, u1,...,u4) = 1 c√µeiθφ(u2,...,u4), e−iθ(pµ−µ3−c2−iµ3 2), (8.1) 20 B.-Y. CHEN, A. PRIETO-MART´ IN, AND X. WANG where cis a positive number, φ:N4→H9 1(−1) ⊂C5 1is a horizontal lift of a non-totally geodesic δ(2)-ideal minimal Lagrangian immersion in CH4(−4), and θ(t)satisfies dθ dµ =1 2p1−µ2−c2µ−1; (iv) a Lagrangian submanifold defined by L(µ, u1,...,u4) = 1 ce−iθ(pµ−µ3+c2−iµ3 2),√µeiθφ(u2,...,u4), (8.2) where cis a positive number, φ:N4→S9(1) ⊂C5is a horizontal lift of a non-totally geodesic δ(2)-ideal minimal Lagrangian immersion in CP4(4), and θ(t)satisfies dθ dµ =1 2p1−µ2+c2µ−1; (v) a Lagrangian submanifold defined by L(t, u1,...,u4) = 1 cosh t−i sinh t2t+w+ icosh 2t−hψ, ψi− 1 4, ψ, 2t+w+ icosh 2t−hψ, ψi+1 4, (8.3) where ψ(u1,...,u4)is a non-totally geodesic δ(2)-ideal Lagrangian minimal immersion in C4and up to a constant w(u1,...,u4)is the unique solution of the PDE system: wuj= 2 ψuj,iψ, j = 1,2,3,4; (vi) a Lagrangian submanifold defined by L(t, u1,...,u4) = 1 cosh t−i sinh t2t+w+ icosh 2t−hψ, ψi− 1 4, ψ1, ψ2,2t+w+ icosh 2t−hψ, ψi+1 4, (8.4) where ψ= (ψ1, ψ2)is the direct product immersion of two non-totally geodesic Lagrangian minimal immersions ψα:N2 α→C2, α = 1,2, and up to a constant w(u1,...,u4)is the unique solution of the PDE system: wuj= 2 ψuj,iψ, j = 1,2,3,4. Proof. Under the hypothesis there exists an orthonormal frame {e1,...,e5} such that (4.1) holds. Case (1) µ= 0. In this case, we obtain case (i) of the theorem. Case (2): µ6= 0 and a, b = 0. In this case Mis an H-umbilical Lagrangian submanifold with ratio 4, which gives case (ii). Case (3): µ6= 0 and at least one of a, b is nonzero. Without loss of generality, we may assume a6= 0 and µ > 0. We divide this into two cases. LAGRANGIAN SUBMANIFOLDS 21 Case (3.a): a, µ 6= 0 and b= 0. By Lemmas 5.1 we obtain (5.1) and (5.2). Also, Mis locally a warped product I×ρ(t)N4with e5=∂taccording to Lemma 5.3. From Lemma 4.1 we find h(e1, e1) = aJe1+µJe5, h(e1, e2) = −aJe2, h(e2, e2) = −aJe1+µJe5, h(e3, e3) = h(e4, e4) = µJe5, h(e5, e5) = 4µJe5, h(ei, e5) = µJei, i ∈∆, h(ei, ej) = 0, otherwise. (8.5) Let Lbe a horizontal immersion of Min H11 1(−1) ⊂C6 1of the Lagrangian immersion of Min CH5(−4) via Hopf ’s fibration and θ(t) a function satisfying dθ dt =µ. (8.6) From Lemma 5.4 we obtain the following ODE system: dν dt = 1 −3µ2−ν2,dµ dt = 2µν. (8.7) It follows from (8.6) and (8.7) that dν dµ =1−3µ2−ν2 2µν ,dθ dµ =1 2ν.(8.8) Solving the first differential equation in (8.8) gives ν=±p1−µ2−kµ−1 for some real number k. By replacing e5by −e5if necessary, we find ν=p1−µ2−kµ−1,dθ dµ =1 2p1−µ2−kµ−1.(8.9) It follows from (8.7) that d dt (1 −µ2−ν2) = −2ν(1 −µ2−ν2).Since this equation for y(t) = 1 −µ2−ν2=kµ−1has a unique solution for each given initial condition, each solution either vanishes identically or is nowhere zero. Case (3.a.1): µ2+ν2<1. In this case, (8.9) implies k > 0. Thus we may put k=c2, c > 0. Consider the maps: η=e−3iθ(e5−(ν+ iµ)L p1−µ2−ν2, φ =e−iθ((ν−iµ)e5−L) p1−µ2−ν2.(8.10) Then hη, ηi= 1 and hφ, φi=−1. From ∇eje5=νej,j∈∆,and (8.5), we obtain ˜ ∇ejξ= 0, where ˜ ∇is the Levi-Civita connection of C6 1. Lemma 5.1 and (8.5) give ˜ ∇e5e5= 4iµe5+L. Thus we find ˜ ∇e5ξ= 0. So ηis a constant 22 B.-Y. CHEN, A. PRIETO-MART´ IN, AND X. WANG unit vector. Also, we find ˜ ∇e5φ= 0. Hence φis independent of t. From (8.10) we get L=−eiθφ+e−iθ(ν−iµ)η p1−µ2−ν2.(8.11) Since φis orthogonal to η, iηand ηis a constant unit space-like vector, we conclude from (8.9) and (8.11) that Lis congruent to (8.1). Next, by applying the same method of the proof of Theorem 7.1, we conclude that φis a horizontal immersion in H9 1(−1) whose second fundamental form is a scalar multiple of the original second fundamental form restricted to the second factor of I×ρN. Consequently, φis a minimal horizontal immersion in H9 1(−1) of a non-totally geodesic δ(2)-ideal Lagrangian minimal submanifold of CH4(−4). This gives case (iii). Case (3.a.2): µ2+ν2>1. In this case (8.8) implies k < 0. Thus we may put k=−c2, c > 0. Now, we consider the maps: η=e−3iθ(e5−(ν+ iµ)L pµ2+ν2−1, φ =e−iθ((ν−iµ)e5−L) pµ2+ν2−1(8.12) instead. Then hφ, φi=−hη, ηi= 1. By applying similar arguments as case (3.a.1), we know that ηis a constant time-like vector and φis independent of tand orthogonal to η, iη. Moreover, we may prove that φis a minimal Legendre immersion in S9(1). Therefore we have case (iv) after choosing η= (1,0,...,0). Case (3.a.3): µ2+ν2= 1. In this case system (8.7) gives dν dt = 2(ν2−1) and µ=±√1−ν2.Solving these and applying a suitable translations in t, we find µ= sech 2t, ν =−tanh 2t. (8.13) It follows from ∇e5e5= 0, (8.5) and (8.13) that the horizontal lift Lof the Lagrangian immersion of Min CH5(−4) ⊂C6 1satisfies Ltt −4i( sech 2t)Lt−L= 0.(8.14) Solving this second order differential equation gives L=φ(u1,...,u4) + B(u1,...,u4)(2t+ i cosh 2t) cosh t−i sinh t,(8.15) where φ(u1,...,u4) and B(u1,...,u4) are C6 1-valued functions. LAGRANGIAN SUBMANIFOLDS 23 On the other hand, it follows from Lemma 5.1, (8.5) and (8.13) that Ltuj= (i sech 2t−tanh 2t)Luj, j ∈∆.(8.16) Substituting (8.15) into (8.16) shows that Bis a constant vector ζ. Thus L(t, u1,...,u4) = φ(u1,...,u4) cosh t−i sinh t+(2t+ i cosh 2t) cosh t−i sinh tζ, (8.17) Since hL, Li=−1, (8.17) implies −cosh 2t=hφ, φi+hφ, (4t+ 2i cosh 2t)ζi+ (4t2+ cosh2(2t)) hζ, ζi.(8.18) Since φt= 0, by differentiating (8.18) with respect twe find −sinh 2t= 2thφ, ζi+ 2 sinh 2thφ, iζi+ (4t+ sinh 4t)hζ, ζi.(8.19) We find from (8.19) at t= 0 that hφ, ζi= 0.Thus (8.19) gives 0 = sinh 2t(1 + hφ, iζi) + (4t+ sinh 4t)hζ, ζi.(8.20) Differentiating (8.20) gives hφ, iζi=−1 2−2hζ, ζi.Thus (8.17) yields hφ, iζi= −1 2and hζ, ζi= 0. Now, we find from (8.18) that hφ, φi= 0. Consequently we have hφ, φi=hζ, ζi=hφ, ζi= 0,hφ, iζi=−1 2.(8.21) Since ζis a constant light-like vector, we may put ζ= (1,0,...,0,1), φ = (a1+ ib1,...,a6+ ib6).(8.22) It follows from (8.21) and (8.22) that a6=a1and b6=b1+1 2. Therefore φ=a1+ ib1, a2+ ib2,...,a1+ i(b1+1 2).(8.23) Now, by using hφ, φi= 0 and (8.23), we find ψ= (a2+ ib2,...,a5+ ib5) and b1=−1 4−hψ, ψi. Combining these with (8.23) yields φ=w−ihψ, ψi− i 4, ψ, w −ihψ, ψi+i 4(8.24) with w=a1. It follows from (8.22) and (8.24) that φuj, ζ=φuj,iζ= 0. Thus, by applying Luj,iL= 0, j ∈∆, we find from (8.17) that φuj,iφ= 0. On the other hand, (8.24) implies that φuj,iφ=−1 2wuj+ψuj,iψ(8.25) with wuj=∂w ∂uj. Therefore wsatisfies the PDE system: wuj= 2 ψuj,iψ. 24 B.-Y. CHEN, A. PRIETO-MART´ IN, AND X. WANG Now, we derive from (8.17), (8.22) and (8.23) that L=1 cosh t−i sinh t2t+w+ i cosh 2t−hψ, ψi− 1 4, ψ, 2t+w+ i cosh 2t−hψ, ψi+1 4. (8.26) It follows from (8.26) that Luj=1 cosh t−i sinh twuj−ihψ, ψiuj, ψuj, wuj−ihψ, ψiuj.(8.27) Thus we find ψuj, ψuk= cosh 2tLuj, Lukwhich implies that ψis an immersion in C4. Also, we find from (8.27) and Luj,iLuk= 0 that ψuj,iψuk= 0. Thus ψis a Lagrangian immersion. Now, by applying an argument similar to the last part of the proof of [11, Theorem 6.1], we conclude that ψujuk= 4 X i=1 (Γi jk + ihi jk)φui, j, k ∈∆. Therefore, according to (8.5), ψis a δ(2)-ideal minimal Lagrangian immersion in C4. Consequently, we obtain case (v) of the theorem. Case (3.b): a, b, µ 6= 0. We obtain case (vi) of the theorem by applying the same argument as case (3.a.3).  Acknowledgement. The authors thank the referee and Dr. Luc Vrancken for pointing out an error in the original version of this paper. References [1] I. Castro and B.-Y. Chen, Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves, Tohoku Math. J. 58 (2006), 565–579. [2] B.-Y. 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Ann.241 (1979), 209–215. [16] H. Reckziegel, Horizontal lifts of isometric immersions into the bundle space of a pseudo-Riemannian submersion, Lecuture Notes in Math. 1156 (1985), 264–279. Department of Mathematics, Michigan State University, East Lansing, Michigan 48824–1027, U.S.A. E-mail address:[email protected] Department of Geometry and Topology, University of Seville, Apdo. de Correos 1160, 41080 - Sevilla, Spain E-mail address:aliciap[email protected] School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P. R. China E-mail address:[email protected]