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Some explicit examples of Hamiltonian minimal Lagrangian submanifolds in complex space forms
Nonlinear Analysis 66 (2007) 1091–1099 www.elsevier.com/locate/na
Some explicit examples of Hamiltonian minimal Lagrangian submanifolds in complex space forms✩ Yuxin Dong ∗ , Yingbo Han Institute of Mathematics, Fudan University, Shanghai 200433, PR China Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, PR China Received 25 November 2005; accepted 5 January 2006
Abstract In this paper, we find some new explicit examples of Hamiltonian minimal Lagrangian submanifolds among the Lagrangian isometric immersions of a real space form in a complex space form. c 2006 Elsevier Ltd. All rights reserved. MSC: primary 05C42, 53D12 Keywords: Real space form; Complex space form; Hamiltonian minimal submanifold
1. Introduction Minimal Lagrangian submanifolds in a Kaehler manifold have received much attention recently, due to their importance in string theory (see [11,3] and the references therein). In [13,14], a nice generalization of minimal Lagrangian submanifolds called the Hamiltonian minimal submanifolds was introduced and investigated by Y.G. Oh. The motivations for studying them were their interesting geometric properties and similarities to some models in incompressible elasticity. A Hamiltonian minimal Lagrangian submanifold will be simply called an H -minimal Lagrangian submanifold. In [4,5,7–10,12], the authors made some efforts to construct examples of H -minimal Lagrangian submanifolds in Kaehler manifolds, especially in complex space forms, by means of symmetry reduction or integrable systems. ✩ Supported by the Zhongdian grant of NSFC. ∗ Corresponding author at: Institute of Mathematics, Fudan University, Shanghai 200433, PR China. Tel.: +86 21 65
64 30 47. E-mail addresses: [email protected] (Y. Dong), [email protected] (Y. Han). c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.01.007
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Y. Dong, Y. Han / Nonlinear Analysis 66 (2007) 1091–1099
In [13], Oh investigated mainly the minimal Lagrangian immersions of R P n and T n in C P n . We note that R P n and T n have constant curvature 1 and 0 respectively. The first of these is actually a totally geodesic Lagrangian immersion. From [1] and [6], we know that a minimal n (4c) ˜ Lagrangian submanifold with constant sectional curvature c in a complex space form M has to be totally geodesic (c = c) ˜ or flat (c = 0). In [2], the authors developed an effective method called twistor product decomposition for constructing Lagrangian isometric immersions n (4c). Later, Y.M. Oh [15] followed of a real space form M n (c) into a complex space form M their method in constructing a lot of examples of such Lagrangian isometric immersions. In this note, we derive a criterion for Hamiltonian minimality for the Lagrangian isometric immersions obtained by the method of twistor product decomposition. Using this criterion, we will find a lot of new explicit H -minimal Lagrangian submanifolds which are not minimal in the usual sense. 2. Preliminaries n , ω) be a complex n-dimensional Kaehler manifold with Kaehler form ω. Let M n Let ( M n . A normal vector field V along M is called a Hamiltonian be a Lagrangian submanifold in M variation if the 1-form αV = ω(V, ·) is exact. According to [13,14], the Lagrangian submanifold M is called H -minimal if it is a critical point of the volume functional with respect to all Hamiltonian variations along M. n , ω, g) be a Kaehler manifold. A Lagrangian submanifold Proposition 2.1 ([14]). Let ( M n M ⊂ M is H -minimal if and only if its mean curvature vector H satisfies δα H = 0
(1)
on M, where δ is the Hodge-dual operator of d on M. We now briefly present the work of [2,15]. Definition 2.2. Let (M1 , g1 ), . . . , (Mm , gm ) be m Riemannian manifolds, f i a positive function on M1 ×· · · × Mm and πi : M1 ×· · · × Mm → Mi the i -th canonical projection for i = 1, . . . , m. The twisted product f1
M1 × · · · ×
fm
Mm
of (M1 , g1 ), . . . , (Mm , gm ) is the differentiable manifold M1 × · · · × Mm equipped with the twisted product metric g defined by g(X, Y ) = f 1 g1 (π1∗ X, π1∗ Y ) + · · · + f m gm (πm∗ X, πm∗ Y ) for all vector fields X and Y of M1 × · · · × Mm . Let N n−k (c) denote an (n − k)-dimensional real space form of constant sectional curvature c. For k < n − 1, we consider the twisted product f 1 I1
× ···×
f k Ik
× N n−k (c)
(2)
with the twisted product metric defined by g = f 1 d x 12 + · · · + f k d x k2 + g0
(3)
where g0 denotes the canonical metric on N n−k (c) and I1 · · · Ik are open intervals. If the twisted product given by (2) is a real space form M n (c) of constant sectional curvature c, then it is called
Y. Dong, Y. Han / Nonlinear Analysis 66 (2007) 1091–1099
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a twisted product decomposition of M n (c). For simplicity, we denote such a decomposition of M n (c) by T P nf1 ··· f k (c). We choose coordinates {x 1 · · · x n } on T P nf1 ··· fk (c) such that ∂ ∂x j is tangent to I j for j = 1, . . . , k and the last n − k coordinate vectors are tangent to N n−k (c). Such coordinates are called adapted coordinates. In [2] the authors introduced the following 1-form Φ(T P) on T P nf1 ··· fk (c): Φ(T P) = f 1 d x 1 + · · · + f k d x k ,
(4)
which is called the twistor form of T P nf1 ··· fk (c). The twistor form Φ(T P) is said to be twisted n (4c) the complex space form closed if ki, j =1 ∂∂xfij d x i ∧ d x j = 0. In this note, we denote by M of constant holomorphic sectional curvature 4c. Proposition 2.3 ([2]). Let T P n (c) = f1 I × · · · × fk I × N n−k (c), for 1 ≤ k ≤ n, be a twisted product decomposition of a simply connected real space form M n (c) with the metric g given by n (4c), (3). If the twistor form Φ of T P nf1 ... fk is twisted closed, then, up to rigid motions of M there is a unique Lagrangian isometric immersion: L f1 ...
fk
: T P nf1 ...
fk
n (4c) →M
whose second fundamental form satisfies ∂ ∂ ∂ h , , j = 1, . . . , k =J ∂x j ∂x j ∂x j ∂ ∂ = 0, otherwise , h ∂ xr ∂ x s for any adapted coordinate system x 1 · · · x n on T P nf1 ··· fk (c). Chen et al. in [2] and Y.M. Oh in [15] considered the following special types of twistor product decomposition: Type 1: f 1 = · · · = f k = f . Type 2: None of f i, s is the same, and ∂∂xfij = 0 for 1 ≤ i = j ≤ k. Type 3: The mixed case, of Type 1 and Type 2. Then they found many Lagrangian isometric immersions by means of Proposition 2.3. Since Type 3 is only the mixed case, of Type 1 and Type 2, we will mainly consider the cases of Type 1 and Type 2. In following, we list partial examples of Type 1 and Type 2 constructed in [2, 15], among which we will find some non-trivial H -minimal Lagrangian immersions in the next section. Example 1. We have the following adapted Lagrangian isometric immersion L f1 ... n (4c): T P nf1 ... f k → M Type 1: (a) c = 0 and k = n = 2: f 1 = f 2 = α 2 e2β(x+y) where α, β are constants, and 1 L(x, y) = √ eζ u (cos(|ζ |v), sin(|ζ |v)) 2|ζ |
fk
:
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Y. Dong, Y. Han / Nonlinear Analysis 66 (2007) 1091–1099
where ζ = β + 2i . (b) c = 0, k = 2 and n ≥ 3: f 1 = f 2 = β(x 1 + x 2 ) 1 +
n
2 cl xl
,
l=3
where β(u) is a positive real valued function, cl are constants, and n n L(x 1 · · · x n ) = A(x 1 , x 2 ) 1 + cl xl + Vl xl l=3
l=3
where Vl are constant vectors in and A(x 1 , x 2 ) is a C n -valued function. (c) c = 1 and k = n = 2: c1 4 (x + y) , f 1 = f 2 = c1 sec 2 Cn
where c1 is a positive constant, and L f1 f2 = π ◦ L where π : S 5 (1) → C P 2 (4) is the Hopf fibration and L is a map into S 5 (1) defined by 1 1 1 1 ci + tan u, e 2 ciu sec u cos(αv), e 2 ciu sec u sin(αv) L= α 2 where u = x + y, v = x − y, α 2 = 1 + (d) c = 1, k = 2 and n ≥ 3: 1
f1 = f2 =
2
cosh
c0 2u
c2 4
and c is a nonzero real number.
(sin x 3 + c1 sin x 4 cos x 3 + · · · + cn−2 cos x n · · · cos x 3 )2 ,
2 , V ···V n+1 , and where c0 = 1 + c12 + · · · + cn−2 0 n−1 are constant vectors in C
L f1 f2 = π ◦ L where π : S 2n+1 → C P n (4) is the Hopf fibration and L is given by L = B0 (x 1 , x 2 )(sin x 3 + c1 sin x 4 cos x 3 + · · · + cn−2 cos x n · · · cos x 3 ) + (V0 sin x 3 + V1 c1 sin x 4 cos x 3 + · · · + Vn−1 cn−2 cos x n · · · cos x 3 ). (e) c = −1 and k = n = 2: Let π H : H 5(−1) → C H 2(−4) be the natural projection. Then the corresponding Lagrangian immersion L f1 ··· fk = π ◦ L m , where L m is an immersion into H 5(−1) (m = 1, . . . , 4) given by: 1 1 1 1 ci − tan u, e 2 ciu sec u cos(αv), e 2 ciu sec u sin(αv) L 1 (x, y) = α 2 where α 2 =
c2 4
− 1 > 0; 1 1 1 1 L 2 (x, y) = e 2 ciu sec u cosh(αv), ci − tan u, e 2 ciu sec u sinh(αv) α 2
Y. Dong, Y. Han / Nonlinear Analysis 66 (2007) 1091–1099
where α 2 = 1 −
1095
c2 4
> 0; 3 1 2 1 −2iu 1 1 1 + v + e L 3 (x, y) = eiu sec u − iu , ieiu sec u − + v 2 4 2 4 2 4 2 1 −2iu 1 + e − iu , veiu sec u ; 4 2 1 1 1 1 L 4 (x, y) = ci + coth u, e 2 ciu arccos u cos(αv), e 2 ciu arccos u sin(αv) α 2 2
where α 2 = 1 + c4 and c > 0. (f) c = −1 and k < n: L f1 ··· fk = π H ◦ L where π H : H 2n+1(−1) → C H n (−4) is the natural projection and L is given by L = A0 [cosh x k+1 + sinh x k+1 (c1 sin x k+2 + · · · + cn−k cos x k+2 · · · cos x n )] + sinh x k+1 (V1 c1 sin x k+2 + · · · + Vn−k cn−k cos x k+2 · · · cos x n ) where A0 is a C1n+1 -valued function of the variables x 1 , . . . , x k , V1 · · · Vn−k are constant vectors in C1n+1 . The twistor functions are f 1 = · · · = f k = a 2 (u)[cosh x k+1 + sinh x k+1 (c1 sin x k+2 + · · · + cn−k cos x k+2 · · · cos x n )]2 where u = x 1 + · · · + x k and c j ( j = 1, . . . , n − k) are constant. Type 2: (g) c = 0: The immersion L f1 ··· fk into C n is given by x1 xk b1 (x)z 1 (x)dx + · · · + bk (x)z k (x)dx L(x 1 · · · x n ) = + x k+1 (z 1 (x 1 ) + · · · + z k (x k )) +
n k
Ai,α (x i )u α
α=k+2 i=1
xi
ai,α z i (x)dx and the twistor functions are 2 n ai,α x α f i = bi (x i ) + x k+1 +
where Ai,α =
(5)
α=k+2
where bi and ai,α are real valued functions of the variable x i only. Moreover, z(x 1 , . . . , x k ) = z 1 (x 1 ) + · · · + z k (x k ) is a k-th special Legendre translation submanifold in S 2n−1 → C n . (h) c = 1: The immersion L f1 ··· fk = π ◦ L where L is an immersion into S 2n+1 given by L = A0 sin x k+1 + A1 sin x k+2 cos x k+1 + · · · + An−k cos x k+1 · · · cos x n where A0 = z 1 (x 1 ) + · · · + z k (x k ) is a k-th special Legendre translation submanifold in S 2n+1 → C n+1 , and A0 , A1 . . . Ak are some C n+1 -valued functional of the variables x 1 · · · x k . The twistor functions are f i = (sin x k+1 + ci,1 sin x k+2 cos x k+1 + · · · + ci,n−k cos x k+1 · · · cos x n )2
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where ci,1 , . . . , ci,n−k are some real valued functions of x i only (1 ≤ i ≤ k). (i) c = −1 and k < n: L f1 ··· f k = π ◦ L where L is an immersion into H 2n+1 given by L = A0 cosh x k+1 + sinh x k+1 (A1 sin x k+2 cos x k+1 + · · · + An−k cos x k+1 · · · cos x n ) where A1 · · · An−k are C1n+1 -valued functions depending on the variables x 1 · · · x k , A0 = z 1 (x 1 ) + · · · + z k (x k ) is a k-th special Legendre translation submanifold in H 2n+1 → C1n+1 . The twistor functions are f i = (cosh x k+1 + sinh x k+1 (di,1 (x i ) sin x k+2 + di,2 (x i ) sin x k+3 cos x k+2 + · · · + di,n−k (x i ) cos x k+2 · · · cos x n ))2 , where di,1 . . . di,n−k are some real valued functions of x i (1 ≤ i ≤ k). Remark 2.4. The above examples in (a), (c), (e) are given in [2] and those in (b), (d), (f), (g), (h) and (i) are constructed in [15]. 3. Hamiltonian minimality criterion In this section, we first give a Hamiltonian minimality criterion for the Lagrangian immersions n (4c). L f1 ··· fk : T P nf1 ··· fk (c) → M Lemma 3.1. Let L f1 ··· fk : T P nf1 ··· fk (c) → M(4c) be the Lagrangian immersion given in Proposition 2.3. Then the Lagrangian immersion L f1 ··· fk is H -minimal if and only if the twistor functions satisfy k j =1
f j−2
k ∂fj 1 ∂ fi − f −1 f j−1 = 0. ∂x j 2 i, j =1 i ∂x j
(6)
Proof. For the immersion L f1 ··· fk , we derive the mean curvature vector field H from Proposition 2.3 as follows: n n 1 ∂ ij (7) H= g hi j = J f ∂x j i, j =1 j =1 j where (g i j ) is the inverse matrix of (gi j ). Now let Y = ki=1 Y i ∂∂xi + Y0 be any tangent vector of L f1 ··· fk such that g( ∂∂x j , Y0 ) = 0 for 1 ≤ j ≤ k. We have ∂ α H (Y ) = ω(H, Y ) = g J H, Y + Y0 ∂ xi i=1 k k k 1 ∂ ∂ , Yi + Y0 = − d x i (Y ). = −g f ∂ x ∂ xi i j j =1 i=1 i=1
k
i
Y. Dong, Y. Han / Nonlinear Analysis 66 (2007) 1091–1099
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It follows that αH = −
k
d xi .
(8)
i=1
From Proposition 2.1, we know that L f1 ··· fk is H -minimal if and only if δα H = 0. Let {e j }nj =1 be a local orthonormal frame on L f1 ··· fk with e j = √1 ∂∂x for 1 ≤ j ≤ k. We compute δα H as fj
j
follows: δα H = −
n
=− =−
(∇ei α H )(ei )
i=1 n
g i j (∇
i, j =1 n
gi j ∇
i, j =1
ij
i, j =1
∂ ∂x j
∂ ∂ xi
g αH ∇
n
=
∂ ∂x j
αH ) αH
∂ ∂ xi
∂ ∂x j
∂ ∂x j
− αH ∇
∂ ∂ xi
∂ ∂x j
that is, n
δα H =
g i j Γilj
(9)
i, j,l=1
where Γilj denotes the Christoffel symbol. By a direct computation, we have
∂gi j 1 lm ∂gm j ∂gim l Γi j = g + − 2 ∂ xi ∂x j ∂ xm 1 −1 ∂ fl ∂ fl = fi δil + δ jl − Ai j l 2 ∂x j ∂ xi where Ai j l =
⎧ ⎨0
∂ fi ⎩δi j ∂ xl
for i or j > k, if i, j ≤ k.
(10)
From (9) and (10), we get δα H =
k j =1
f j−1
k ∂fj ∂ fi 1 − f −1 f j−1 . ∂x j 2 i, j =1 i ∂x j
Hence the Lagrangian immersion L f1 ··· fk is H -minimal if and only if k j =1
f j−2
k ∂fj ∂ fi 1 − f −1 f j−1 = 0. ∂x j 2 i, j =1 i ∂x j
It is difficult to solve the Eq. (6) in general. But we may still find many interesting solutions of Eq. (6) in some special cases.
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Y. Dong, Y. Han / Nonlinear Analysis 66 (2007) 1091–1099
n (4c) with k = 2 Theorem 3.2. Any type-1 Lagrangian immersion L f1 ··· fk : T P nf1 ··· f k (c) → M is Hamiltonian minimal. Proof. By the assumptions that f 1 = f 2 = f and k = 2, we verify Eq. (6) as follows: k j =1
f j−2
k 2 2 ∂fj ∂fj ∂ fi ∂ fi 1 1 − f i−1 f j−1 = f j−2 − f i−1 f j−1 ∂x j 2 i, j =1 ∂x j ∂x j 2 i, j =1 ∂x j j =1
= 0. Therefore (a), (b), (c), (d) and (e) of Example 1 are H -minimal Lagrangian submanifolds. For the case (f), we already see that it is H -minimal when k = 2. If k > 2, it is easy to derive that the Lagrangian immersion (f) is H -minimal if and only if the function a(u) in (f) is constant. Let us now consider the case (g). Put (5) into (6); then the H -minimal condition for (g) becomes k
bi +
i=1
n α=k+2
bi + x k+1 +
x ai,α α
n α=k+2
3 = 0.
(11)
ai,α x α
We see that (11) holds provided that ai,α , bi are all constant. For the cases (h) and (i), we may show similarly that: if {ci,1 , . . . , ci,n−k } or {di,1 , . . . , di,n−k } are constant, then the corresponding Lagrangian immersions L f1 ··· fk satisfy the Hamiltonian minimal equation (6). In conclusion, we have shown the following: n (4c) be the Lagrangian immersion Theorem 3.3. Let L f1 ··· fk : T P nf1 ··· fk (c) → M in (f),(g),(h) or (i) of Example 1. If the functions a(u), {bi , ai,α }, {ci, j } or {di, j } are constant, then the corresponding Lagrangian immersions are H -minimal. From Theorem 2.9 of [5], we know that the inverse image of an H -minimal Lagrangian submanifold in C P n under the Hopf projection is also H -minimal in C n+1 . Hence we may obtain further H -minimal Lagrangian submanifolds in C n+1 by taking the inverse image of those H -minimal Lagrangian submanifolds found in C P n by the above method. References [1] B.Y. Chen, K. Oguie, On totally real submanifolds, Trans. Amer. Math. Soc. 193 (1974) 257–266. [2] B.Y. Chen, F. Dillen, L. Verstraelen, L. Vrancken, Lagrangian isometric immersions of a real-space-form M n (n) into a complex-space-form M(4c), Math. Proc. Cambridge Philos. Soc. 124 (1998) 107–125. [3] B.Y. Chen, Riemannian geometry of Lagrangian submanifolds, Taiwanese J. Math. 5 (4) (2001) 681–723. [4] I. Castro, F. Urbano, Example of unstable Hamiltonian minimal Lagrangian tori in C P 2 , Compos. Math. 111 (1998) 1–14. [5] Y.X. Dong, Hamiltonian minimal submanifolds in Kaehler manifolds with symmetries, arXiv:math.DG/0412229v1, 12 December 2004. [6] N. Ejiri, Totally real minimal immersions of n-dimensional real space forms into n-dimensional complex space forms, Proc. Amer. Math. Soc. 84 (1982) 117–123. [7] F. Helein, P. Romon, Hamiltonian stationary Lagrangian surfaces in C 2 , Comm. Anal. Geom. 10 (1) (2002) 79–126. [8] F. Helein, P Romon, Hamiltonian stationary tori in the complex projective plane, Proc. London Math. Soc. (2) 90 (2005) 472–496.
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[9] I. Castro, H.Z. Li, F. Urbano, Hamiltonian-minimal Lagrangian submanifolds in complex space forms, arXiv:math.DG/0412046v1, 2 December 2004. [10] Q.C. Ji, Hamiltonian minimal submanifolds in complex hyperbolic space with S O(n)-symmetry (in press). [11] D. Joyce, Lectures on a Calabi–Yau and special Lagrangian geometry, arXiv:math.DG0108088v3, 25 June 2002. [12] A. Mironov, On new examples of Hamiltonian-minimal and minimal Lagrangian submanifolds in C n and C P n , arXiv:math.DG/0309128v1, 2003. [13] Y.G. Oh, Second variation and stabilities of minimal Lagrangian submanifolds in Kaehler manifolds, Invent. Math. 101 (1990) 501–519. [14] Y.G. Oh, Volume minimization of Lagrangian submanifolds under Hamiltonian deformations, Math. Z. 212 (1993) 175–192. [15] Y.M. Oh, Explicit construction of Lagrangian isometric immersion of a real space form M n (c) into a complex space n (4c), Math. Proc. Cambridge Philos. Soc. 132 (3) (2002) 481–508. form M
Some explicit examples of Hamiltonian minimal Lagrangian submanifolds in complex space forms✩ Yuxin Dong ∗ , Yingbo Han Institute of Mathematics, Fudan University, Shanghai 200433, PR China Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, PR China Received 25 November 2005; accepted 5 January 2006
Abstract In this paper, we find some new explicit examples of Hamiltonian minimal Lagrangian submanifolds among the Lagrangian isometric immersions of a real space form in a complex space form. c 2006 Elsevier Ltd. All rights reserved. MSC: primary 05C42, 53D12 Keywords: Real space form; Complex space form; Hamiltonian minimal submanifold
1. Introduction Minimal Lagrangian submanifolds in a Kaehler manifold have received much attention recently, due to their importance in string theory (see [11,3] and the references therein). In [13,14], a nice generalization of minimal Lagrangian submanifolds called the Hamiltonian minimal submanifolds was introduced and investigated by Y.G. Oh. The motivations for studying them were their interesting geometric properties and similarities to some models in incompressible elasticity. A Hamiltonian minimal Lagrangian submanifold will be simply called an H -minimal Lagrangian submanifold. In [4,5,7–10,12], the authors made some efforts to construct examples of H -minimal Lagrangian submanifolds in Kaehler manifolds, especially in complex space forms, by means of symmetry reduction or integrable systems. ✩ Supported by the Zhongdian grant of NSFC. ∗ Corresponding author at: Institute of Mathematics, Fudan University, Shanghai 200433, PR China. Tel.: +86 21 65
64 30 47. E-mail addresses: [email protected] (Y. Dong), [email protected] (Y. Han). c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.01.007
1092
Y. Dong, Y. Han / Nonlinear Analysis 66 (2007) 1091–1099
In [13], Oh investigated mainly the minimal Lagrangian immersions of R P n and T n in C P n . We note that R P n and T n have constant curvature 1 and 0 respectively. The first of these is actually a totally geodesic Lagrangian immersion. From [1] and [6], we know that a minimal n (4c) ˜ Lagrangian submanifold with constant sectional curvature c in a complex space form M has to be totally geodesic (c = c) ˜ or flat (c = 0). In [2], the authors developed an effective method called twistor product decomposition for constructing Lagrangian isometric immersions n (4c). Later, Y.M. Oh [15] followed of a real space form M n (c) into a complex space form M their method in constructing a lot of examples of such Lagrangian isometric immersions. In this note, we derive a criterion for Hamiltonian minimality for the Lagrangian isometric immersions obtained by the method of twistor product decomposition. Using this criterion, we will find a lot of new explicit H -minimal Lagrangian submanifolds which are not minimal in the usual sense. 2. Preliminaries n , ω) be a complex n-dimensional Kaehler manifold with Kaehler form ω. Let M n Let ( M n . A normal vector field V along M is called a Hamiltonian be a Lagrangian submanifold in M variation if the 1-form αV = ω(V, ·) is exact. According to [13,14], the Lagrangian submanifold M is called H -minimal if it is a critical point of the volume functional with respect to all Hamiltonian variations along M. n , ω, g) be a Kaehler manifold. A Lagrangian submanifold Proposition 2.1 ([14]). Let ( M n M ⊂ M is H -minimal if and only if its mean curvature vector H satisfies δα H = 0
(1)
on M, where δ is the Hodge-dual operator of d on M. We now briefly present the work of [2,15]. Definition 2.2. Let (M1 , g1 ), . . . , (Mm , gm ) be m Riemannian manifolds, f i a positive function on M1 ×· · · × Mm and πi : M1 ×· · · × Mm → Mi the i -th canonical projection for i = 1, . . . , m. The twisted product f1
M1 × · · · ×
fm
Mm
of (M1 , g1 ), . . . , (Mm , gm ) is the differentiable manifold M1 × · · · × Mm equipped with the twisted product metric g defined by g(X, Y ) = f 1 g1 (π1∗ X, π1∗ Y ) + · · · + f m gm (πm∗ X, πm∗ Y ) for all vector fields X and Y of M1 × · · · × Mm . Let N n−k (c) denote an (n − k)-dimensional real space form of constant sectional curvature c. For k < n − 1, we consider the twisted product f 1 I1
× ···×
f k Ik
× N n−k (c)
(2)
with the twisted product metric defined by g = f 1 d x 12 + · · · + f k d x k2 + g0
(3)
where g0 denotes the canonical metric on N n−k (c) and I1 · · · Ik are open intervals. If the twisted product given by (2) is a real space form M n (c) of constant sectional curvature c, then it is called
Y. Dong, Y. Han / Nonlinear Analysis 66 (2007) 1091–1099
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a twisted product decomposition of M n (c). For simplicity, we denote such a decomposition of M n (c) by T P nf1 ··· f k (c). We choose coordinates {x 1 · · · x n } on T P nf1 ··· fk (c) such that ∂ ∂x j is tangent to I j for j = 1, . . . , k and the last n − k coordinate vectors are tangent to N n−k (c). Such coordinates are called adapted coordinates. In [2] the authors introduced the following 1-form Φ(T P) on T P nf1 ··· fk (c): Φ(T P) = f 1 d x 1 + · · · + f k d x k ,
(4)
which is called the twistor form of T P nf1 ··· fk (c). The twistor form Φ(T P) is said to be twisted n (4c) the complex space form closed if ki, j =1 ∂∂xfij d x i ∧ d x j = 0. In this note, we denote by M of constant holomorphic sectional curvature 4c. Proposition 2.3 ([2]). Let T P n (c) = f1 I × · · · × fk I × N n−k (c), for 1 ≤ k ≤ n, be a twisted product decomposition of a simply connected real space form M n (c) with the metric g given by n (4c), (3). If the twistor form Φ of T P nf1 ... fk is twisted closed, then, up to rigid motions of M there is a unique Lagrangian isometric immersion: L f1 ...
fk
: T P nf1 ...
fk
n (4c) →M
whose second fundamental form satisfies ∂ ∂ ∂ h , , j = 1, . . . , k =J ∂x j ∂x j ∂x j ∂ ∂ = 0, otherwise , h ∂ xr ∂ x s for any adapted coordinate system x 1 · · · x n on T P nf1 ··· fk (c). Chen et al. in [2] and Y.M. Oh in [15] considered the following special types of twistor product decomposition: Type 1: f 1 = · · · = f k = f . Type 2: None of f i, s is the same, and ∂∂xfij = 0 for 1 ≤ i = j ≤ k. Type 3: The mixed case, of Type 1 and Type 2. Then they found many Lagrangian isometric immersions by means of Proposition 2.3. Since Type 3 is only the mixed case, of Type 1 and Type 2, we will mainly consider the cases of Type 1 and Type 2. In following, we list partial examples of Type 1 and Type 2 constructed in [2, 15], among which we will find some non-trivial H -minimal Lagrangian immersions in the next section. Example 1. We have the following adapted Lagrangian isometric immersion L f1 ... n (4c): T P nf1 ... f k → M Type 1: (a) c = 0 and k = n = 2: f 1 = f 2 = α 2 e2β(x+y) where α, β are constants, and 1 L(x, y) = √ eζ u (cos(|ζ |v), sin(|ζ |v)) 2|ζ |
fk
:
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Y. Dong, Y. Han / Nonlinear Analysis 66 (2007) 1091–1099
where ζ = β + 2i . (b) c = 0, k = 2 and n ≥ 3: f 1 = f 2 = β(x 1 + x 2 ) 1 +
n
2 cl xl
,
l=3
where β(u) is a positive real valued function, cl are constants, and n n L(x 1 · · · x n ) = A(x 1 , x 2 ) 1 + cl xl + Vl xl l=3
l=3
where Vl are constant vectors in and A(x 1 , x 2 ) is a C n -valued function. (c) c = 1 and k = n = 2: c1 4 (x + y) , f 1 = f 2 = c1 sec 2 Cn
where c1 is a positive constant, and L f1 f2 = π ◦ L where π : S 5 (1) → C P 2 (4) is the Hopf fibration and L is a map into S 5 (1) defined by 1 1 1 1 ci + tan u, e 2 ciu sec u cos(αv), e 2 ciu sec u sin(αv) L= α 2 where u = x + y, v = x − y, α 2 = 1 + (d) c = 1, k = 2 and n ≥ 3: 1
f1 = f2 =
2
cosh
c0 2u
c2 4
and c is a nonzero real number.
(sin x 3 + c1 sin x 4 cos x 3 + · · · + cn−2 cos x n · · · cos x 3 )2 ,
2 , V ···V n+1 , and where c0 = 1 + c12 + · · · + cn−2 0 n−1 are constant vectors in C
L f1 f2 = π ◦ L where π : S 2n+1 → C P n (4) is the Hopf fibration and L is given by L = B0 (x 1 , x 2 )(sin x 3 + c1 sin x 4 cos x 3 + · · · + cn−2 cos x n · · · cos x 3 ) + (V0 sin x 3 + V1 c1 sin x 4 cos x 3 + · · · + Vn−1 cn−2 cos x n · · · cos x 3 ). (e) c = −1 and k = n = 2: Let π H : H 5(−1) → C H 2(−4) be the natural projection. Then the corresponding Lagrangian immersion L f1 ··· fk = π ◦ L m , where L m is an immersion into H 5(−1) (m = 1, . . . , 4) given by: 1 1 1 1 ci − tan u, e 2 ciu sec u cos(αv), e 2 ciu sec u sin(αv) L 1 (x, y) = α 2 where α 2 =
c2 4
− 1 > 0; 1 1 1 1 L 2 (x, y) = e 2 ciu sec u cosh(αv), ci − tan u, e 2 ciu sec u sinh(αv) α 2
Y. Dong, Y. Han / Nonlinear Analysis 66 (2007) 1091–1099
where α 2 = 1 −
1095
c2 4
> 0; 3 1 2 1 −2iu 1 1 1 + v + e L 3 (x, y) = eiu sec u − iu , ieiu sec u − + v 2 4 2 4 2 4 2 1 −2iu 1 + e − iu , veiu sec u ; 4 2 1 1 1 1 L 4 (x, y) = ci + coth u, e 2 ciu arccos u cos(αv), e 2 ciu arccos u sin(αv) α 2 2
where α 2 = 1 + c4 and c > 0. (f) c = −1 and k < n: L f1 ··· fk = π H ◦ L where π H : H 2n+1(−1) → C H n (−4) is the natural projection and L is given by L = A0 [cosh x k+1 + sinh x k+1 (c1 sin x k+2 + · · · + cn−k cos x k+2 · · · cos x n )] + sinh x k+1 (V1 c1 sin x k+2 + · · · + Vn−k cn−k cos x k+2 · · · cos x n ) where A0 is a C1n+1 -valued function of the variables x 1 , . . . , x k , V1 · · · Vn−k are constant vectors in C1n+1 . The twistor functions are f 1 = · · · = f k = a 2 (u)[cosh x k+1 + sinh x k+1 (c1 sin x k+2 + · · · + cn−k cos x k+2 · · · cos x n )]2 where u = x 1 + · · · + x k and c j ( j = 1, . . . , n − k) are constant. Type 2: (g) c = 0: The immersion L f1 ··· fk into C n is given by x1 xk b1 (x)z 1 (x)dx + · · · + bk (x)z k (x)dx L(x 1 · · · x n ) = + x k+1 (z 1 (x 1 ) + · · · + z k (x k )) +
n k
Ai,α (x i )u α
α=k+2 i=1
xi
ai,α z i (x)dx and the twistor functions are 2 n ai,α x α f i = bi (x i ) + x k+1 +
where Ai,α =
(5)
α=k+2
where bi and ai,α are real valued functions of the variable x i only. Moreover, z(x 1 , . . . , x k ) = z 1 (x 1 ) + · · · + z k (x k ) is a k-th special Legendre translation submanifold in S 2n−1 → C n . (h) c = 1: The immersion L f1 ··· fk = π ◦ L where L is an immersion into S 2n+1 given by L = A0 sin x k+1 + A1 sin x k+2 cos x k+1 + · · · + An−k cos x k+1 · · · cos x n where A0 = z 1 (x 1 ) + · · · + z k (x k ) is a k-th special Legendre translation submanifold in S 2n+1 → C n+1 , and A0 , A1 . . . Ak are some C n+1 -valued functional of the variables x 1 · · · x k . The twistor functions are f i = (sin x k+1 + ci,1 sin x k+2 cos x k+1 + · · · + ci,n−k cos x k+1 · · · cos x n )2
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Y. Dong, Y. Han / Nonlinear Analysis 66 (2007) 1091–1099
where ci,1 , . . . , ci,n−k are some real valued functions of x i only (1 ≤ i ≤ k). (i) c = −1 and k < n: L f1 ··· f k = π ◦ L where L is an immersion into H 2n+1 given by L = A0 cosh x k+1 + sinh x k+1 (A1 sin x k+2 cos x k+1 + · · · + An−k cos x k+1 · · · cos x n ) where A1 · · · An−k are C1n+1 -valued functions depending on the variables x 1 · · · x k , A0 = z 1 (x 1 ) + · · · + z k (x k ) is a k-th special Legendre translation submanifold in H 2n+1 → C1n+1 . The twistor functions are f i = (cosh x k+1 + sinh x k+1 (di,1 (x i ) sin x k+2 + di,2 (x i ) sin x k+3 cos x k+2 + · · · + di,n−k (x i ) cos x k+2 · · · cos x n ))2 , where di,1 . . . di,n−k are some real valued functions of x i (1 ≤ i ≤ k). Remark 2.4. The above examples in (a), (c), (e) are given in [2] and those in (b), (d), (f), (g), (h) and (i) are constructed in [15]. 3. Hamiltonian minimality criterion In this section, we first give a Hamiltonian minimality criterion for the Lagrangian immersions n (4c). L f1 ··· fk : T P nf1 ··· fk (c) → M Lemma 3.1. Let L f1 ··· fk : T P nf1 ··· fk (c) → M(4c) be the Lagrangian immersion given in Proposition 2.3. Then the Lagrangian immersion L f1 ··· fk is H -minimal if and only if the twistor functions satisfy k j =1
f j−2
k ∂fj 1 ∂ fi − f −1 f j−1 = 0. ∂x j 2 i, j =1 i ∂x j
(6)
Proof. For the immersion L f1 ··· fk , we derive the mean curvature vector field H from Proposition 2.3 as follows: n n 1 ∂ ij (7) H= g hi j = J f ∂x j i, j =1 j =1 j where (g i j ) is the inverse matrix of (gi j ). Now let Y = ki=1 Y i ∂∂xi + Y0 be any tangent vector of L f1 ··· fk such that g( ∂∂x j , Y0 ) = 0 for 1 ≤ j ≤ k. We have ∂ α H (Y ) = ω(H, Y ) = g J H, Y + Y0 ∂ xi i=1 k k k 1 ∂ ∂ , Yi + Y0 = − d x i (Y ). = −g f ∂ x ∂ xi i j j =1 i=1 i=1
k
i
Y. Dong, Y. Han / Nonlinear Analysis 66 (2007) 1091–1099
1097
It follows that αH = −
k
d xi .
(8)
i=1
From Proposition 2.1, we know that L f1 ··· fk is H -minimal if and only if δα H = 0. Let {e j }nj =1 be a local orthonormal frame on L f1 ··· fk with e j = √1 ∂∂x for 1 ≤ j ≤ k. We compute δα H as fj
j
follows: δα H = −
n
=− =−
(∇ei α H )(ei )
i=1 n
g i j (∇
i, j =1 n
gi j ∇
i, j =1
ij
i, j =1
∂ ∂x j
∂ ∂ xi
g αH ∇
n
=
∂ ∂x j
αH ) αH
∂ ∂ xi
∂ ∂x j
∂ ∂x j
− αH ∇
∂ ∂ xi
∂ ∂x j
that is, n
δα H =
g i j Γilj
(9)
i, j,l=1
where Γilj denotes the Christoffel symbol. By a direct computation, we have
∂gi j 1 lm ∂gm j ∂gim l Γi j = g + − 2 ∂ xi ∂x j ∂ xm 1 −1 ∂ fl ∂ fl = fi δil + δ jl − Ai j l 2 ∂x j ∂ xi where Ai j l =
⎧ ⎨0
∂ fi ⎩δi j ∂ xl
for i or j > k, if i, j ≤ k.
(10)
From (9) and (10), we get δα H =
k j =1
f j−1
k ∂fj ∂ fi 1 − f −1 f j−1 . ∂x j 2 i, j =1 i ∂x j
Hence the Lagrangian immersion L f1 ··· fk is H -minimal if and only if k j =1
f j−2
k ∂fj ∂ fi 1 − f −1 f j−1 = 0. ∂x j 2 i, j =1 i ∂x j
It is difficult to solve the Eq. (6) in general. But we may still find many interesting solutions of Eq. (6) in some special cases.
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Y. Dong, Y. Han / Nonlinear Analysis 66 (2007) 1091–1099
n (4c) with k = 2 Theorem 3.2. Any type-1 Lagrangian immersion L f1 ··· fk : T P nf1 ··· f k (c) → M is Hamiltonian minimal. Proof. By the assumptions that f 1 = f 2 = f and k = 2, we verify Eq. (6) as follows: k j =1
f j−2
k 2 2 ∂fj ∂fj ∂ fi ∂ fi 1 1 − f i−1 f j−1 = f j−2 − f i−1 f j−1 ∂x j 2 i, j =1 ∂x j ∂x j 2 i, j =1 ∂x j j =1
= 0. Therefore (a), (b), (c), (d) and (e) of Example 1 are H -minimal Lagrangian submanifolds. For the case (f), we already see that it is H -minimal when k = 2. If k > 2, it is easy to derive that the Lagrangian immersion (f) is H -minimal if and only if the function a(u) in (f) is constant. Let us now consider the case (g). Put (5) into (6); then the H -minimal condition for (g) becomes k
bi +
i=1
n α=k+2
bi + x k+1 +
x ai,α α
n α=k+2
3 = 0.
(11)
ai,α x α
We see that (11) holds provided that ai,α , bi are all constant. For the cases (h) and (i), we may show similarly that: if {ci,1 , . . . , ci,n−k } or {di,1 , . . . , di,n−k } are constant, then the corresponding Lagrangian immersions L f1 ··· fk satisfy the Hamiltonian minimal equation (6). In conclusion, we have shown the following: n (4c) be the Lagrangian immersion Theorem 3.3. Let L f1 ··· fk : T P nf1 ··· fk (c) → M in (f),(g),(h) or (i) of Example 1. If the functions a(u), {bi , ai,α }, {ci, j } or {di, j } are constant, then the corresponding Lagrangian immersions are H -minimal. From Theorem 2.9 of [5], we know that the inverse image of an H -minimal Lagrangian submanifold in C P n under the Hopf projection is also H -minimal in C n+1 . Hence we may obtain further H -minimal Lagrangian submanifolds in C n+1 by taking the inverse image of those H -minimal Lagrangian submanifolds found in C P n by the above method. References [1] B.Y. Chen, K. Oguie, On totally real submanifolds, Trans. Amer. Math. Soc. 193 (1974) 257–266. [2] B.Y. Chen, F. Dillen, L. Verstraelen, L. Vrancken, Lagrangian isometric immersions of a real-space-form M n (n) into a complex-space-form M(4c), Math. Proc. Cambridge Philos. Soc. 124 (1998) 107–125. [3] B.Y. Chen, Riemannian geometry of Lagrangian submanifolds, Taiwanese J. Math. 5 (4) (2001) 681–723. [4] I. Castro, F. Urbano, Example of unstable Hamiltonian minimal Lagrangian tori in C P 2 , Compos. Math. 111 (1998) 1–14. [5] Y.X. Dong, Hamiltonian minimal submanifolds in Kaehler manifolds with symmetries, arXiv:math.DG/0412229v1, 12 December 2004. [6] N. Ejiri, Totally real minimal immersions of n-dimensional real space forms into n-dimensional complex space forms, Proc. Amer. Math. Soc. 84 (1982) 117–123. [7] F. Helein, P. Romon, Hamiltonian stationary Lagrangian surfaces in C 2 , Comm. Anal. Geom. 10 (1) (2002) 79–126. [8] F. Helein, P Romon, Hamiltonian stationary tori in the complex projective plane, Proc. London Math. Soc. (2) 90 (2005) 472–496.
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[9] I. Castro, H.Z. Li, F. Urbano, Hamiltonian-minimal Lagrangian submanifolds in complex space forms, arXiv:math.DG/0412046v1, 2 December 2004. [10] Q.C. Ji, Hamiltonian minimal submanifolds in complex hyperbolic space with S O(n)-symmetry (in press). [11] D. Joyce, Lectures on a Calabi–Yau and special Lagrangian geometry, arXiv:math.DG0108088v3, 25 June 2002. [12] A. Mironov, On new examples of Hamiltonian-minimal and minimal Lagrangian submanifolds in C n and C P n , arXiv:math.DG/0309128v1, 2003. [13] Y.G. Oh, Second variation and stabilities of minimal Lagrangian submanifolds in Kaehler manifolds, Invent. Math. 101 (1990) 501–519. [14] Y.G. Oh, Volume minimization of Lagrangian submanifolds under Hamiltonian deformations, Math. Z. 212 (1993) 175–192. [15] Y.M. Oh, Explicit construction of Lagrangian isometric immersion of a real space form M n (c) into a complex space n (4c), Math. Proc. Cambridge Philos. Soc. 132 (3) (2002) 481–508. form M