Lagrangian surfaces in the complex hyperquadric Q2

Journal of Geometry and Physics 97 (2015) 61–68

Contents lists available at ScienceDirect

Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/jgp

Lagrangian surfaces in the complex hyperquadric Q2 Jun Wang a , Xiaowei Xu b,c,∗ a

School of Mathematics Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, China

b

School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui province, China

c

Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, Hefei, 230026, Anhui, China

article

info

Article history: Received 25 February 2015 Accepted 3 July 2015 Available online 9 July 2015 MSC: 53C40 53C42 53D12

abstract In this paper, we introduce a one-form Φ associated to a Lagrangian immersion f from Riemann surface M into the complex hyperquadric Q2 . It is proved that f is minimal if and only if Φ is vanishing and f is H-minimal if and only if Φ is holomorphic. As an application, a family of S 1 -equivariant minimal Lagrangian tori in Q2 are obtained. © 2015 Published by Elsevier B.V.

Keywords: Lagrangian surface Minimal surface Holomorphic one-form

1. Introduction Let (N , ω) be a Kähler manifold with complex dimension n, where ω is its Kähler form. An immersion f : M −→ N from a real n-dimensional manifold M into N is said to be Lagrangian if f ∗ ω = 0. For a Lagrangian immersion f , there is a one-form defined by αH = (H ⌋ω)|M , where H is the mean curvature vector. Minimal (resp. Hamiltonian-minimal or H-minimal for short) Lagrangian submanifolds of a Kähler manifold are critical points of the volume functional under variations (resp. on Lagrangian submanifolds under Hamiltonian variations [1]). It is known that f is minimal if and only if αH = 0, and f is H-minimal if and only if δαH = 0, where δ is the codifferential operator on M w.r.t. the induced metric. In the past few decades, many geometricians have constructed minimal or H-minimal Lagrangian submanifolds in the complex space forms. I. Castro, H. Li and F. Urbano [2] used the Legendrian immersions in odd-dimensional spheres and antide Sitter spaces to construct minimal and H-minimal Lagrangian submanifolds in CPn and CHn . I. Castro and F. Urbano [3] gave new examples of minimal Lagrangian tori in CP2 , and in [4] they construct unstable H-minimal Lagrangian tori in C2 . B.Y. Cheng and O.J. Garay [5] classified H-minimal Lagrangian submanifolds with constant curvature in CP3 with positive nullity. R. Chiang [6] gave Lagrangian submanifolds in CPn with interesting topology feature. Y.X. Dong [7] constructed H-minimal Lagrangian submanifolds in Kähler manifold with symmetric by using the moment map. F. Helen and P. Rommon [8,9] studied a general construction of H-minimal Lagrangian surfaces in C2 and CP2 from the point of view of completely integrable systems. H. Ma and her cooperators [10–12] studied the Lagrangian tori in CP2 from different viewpoints. A.E. Mironov [13–17] constructed some examples of H-minimal and minimal Lagrangian submanifolds in Cn and CPn for higher dimensional cases.

∗

Corresponding author at: School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui province, China. E-mail addresses: [email protected] (J. Wang), [email protected] (X. Xu).

http://dx.doi.org/10.1016/j.geomphys.2015.07.009 0393-0440/© 2015 Published by Elsevier B.V.

62

J. Wang, X. Xu / Journal of Geometry and Physics 97 (2015) 61–68

In this paper, we will study the Lagrangian surfaces in the complex hyperquadric Q2 with the natural Kähler structure induced from CP3 . To study the Lagrangian surfaces in Q2 , we introduce a one-form Φ to measure the minimality of Lagrangian surfaces in Q2 . More explicitly, let f : M −→ Q2 be a Lagrangian immersion, then f is minimal if and only if the associated one-form Φ = 0, and f is H-minimal if and only if Φ is holomorphic. Our paper is organized as follows. In Section 2, we recall some basic formulae of surfaces in Q2 . In Section 3, we prove the basic theorem (Theorem 3.2) about the Lagrangian surfaces in Q2 . In Section 4, under some additional conditions, we reduce the PDE-system (the equations in (3.7)) of the existence of Lagrangian surfaces to an ODE-system, and we obtain a family of minimal Lagrangian tori in Q2 by solving this ODE-system. 2. Formulae of surfaces in Q2 In this section, we recall some basic formulae of surfaces in the complex hyperquadric Q2 . Let Cn be the complex Euclidean space. The ‘‘dot’’ product is a symmetric bilinear form which defined by z·w=

n 

zk wk ,

for z = (z1 , . . . , zn ), w = (w1 , . . . , wn ) ∈ Cn .

k=1

¯ where w ¯ is the conjugate of w. Then the standard Hermitian inner product ( , ) on Cn is given by (z, w) = z · w,   Let Q2 = [z] ∈ CP3 | z · z = 0 be the complex hyperquadric. It is a complex algebraic submanifold in the complex projective space CP3 , and it is also isomorphic to the symmetric space O(4)/(SO(2) × O(2)). A unitary frame of C4 is an ordered set of 4 linearly independent vectors e0 , e1 , e2 , e3 such that (eA , eB ) = δAB ,

0 ≤ A, B, C , . . . ≤ 3.

(2.1)

Writing deA =



ωAB¯ eB ,

(2.2)

B

where ωAB¯ = (deA , eB ) are the Maurer–Cartan forms of U (4). They are skew-Hermitian, i.e.,

ωAB¯ + ωBA ¯ = 0,

(2.3)

where ωBA ¯ is the conjugate of ωBA¯ . Taking the exterior derivative of (2.2), we obtain the Maurer–Cartan equations dωAB¯ =



ωAC¯ ∧ ωC B¯ .

(2.4)

c

Locally, we say a moving frame of Q2 is a unitary frame of C4 which satisfying e3 = e¯ 0 .

(2.5)

By using (2.2) and (2.5), we obtain

ω03¯ = 0.

(2.6) 3

Hence, the metric on Q2 induced from the Fubini-Study metric of CP is given by ds2 = ω01¯ ω01 ¯ + ω02¯ ω02 ¯ ,

(2.7)

and the Kähler form is given by

ω=

i 2

(ω01¯ ∧ ω01 ¯ + ω02¯ ∧ ω02 ¯ ).

(2.8)

Let (M , ds2M ) be a connected Riemann surface with the prescribed metric ds2M . Locally, we choose a complex-valued oneform θ , which is defined up to a complex factor of absolute value one, such that the metric is given by ds2M = θ θ¯ .

(2.9)

The first structure equation of

ds2M

is

dθ = ρ ∧ θ ,

(2.10)

where the purely imaginary one-form ρ is the connection form of ds2M . Let f : M −→ Q2 be a conformal isometric immersion from M into the complex hyperquadric Q2 . Set f ∗ ω0k¯ = ak θ + bk θ¯ ,

k = 1, 2.

(2.11)

Taking the exterior derivative on both sides of (2.11), by using (2.4), (2.6) and (2.10), we obtain Dak ∧ θ + Dbk ∧ θ¯ = 0,

(2.12)

J. Wang, X. Xu / Journal of Geometry and Physics 97 (2015) 61–68

63

where Dak , Dbk are the covariant differential of ak , bk , which are defined by Dak = dak + ak ρ + al ωlk¯ ,

Dbk = dbk − bk ρ + bl ωlk¯ .

(2.13)

According to (2.12), we can write Dak = pk θ + qk θ¯ ,

Dbk = qk θ + rk θ¯ .

(2.14)

It is known (see [18]) that f is minimal if and only if for k = 1, 2.

qk = 0

(2.15)

Denote the mean curvature vector of f by H. Then the associated one-form αH = H ⌋ω of f can be written as

αH = (h1 − h2 )θ + (h1 + h2 )θ¯ ,

(2.16)

where h1 = i

2  [qk (ak¯ + bk¯ ) − qk¯ (ak + bk )],

h2 = i

k=1

2  [qk (ak¯ − bk¯ ) + qk¯ (ak − bk )].

(2.17)

k=1

Define the covariant differential of hk by Dh1 = dh1 − h2 ρ,

Dh2 = dh2 − h1 ρ,

(2.18)

and writing Dh1 = pθ + qθ¯ ,

Dh2 = r θ + sθ¯ .

(2.19)

Base on the relations between the real and complex second fundamental form (see [19] for details), it is easy to check that δαH = 0 if and only if p + q + r − s = 0,

(2.20)

where δ the codifferential operator w.r.t. the metric

ds2M .

3. Lagrangian surfaces in Q2 In this section, we prove a basic theorem of Lagrangian surfaces in the complex hyperquadric Q2 . In this section, we will view Q2 as a submanifold in CP3 . Let f : M −→ Q2 be a Lagrangian conformal isometric immersion, U ⊂ M be a simply connected domain with holomorphic coordinate z = x + iy. Without loss of generality, the metric on M can be written as ds2M = 2eu dzdz¯ . Locally, there is a horizontal lifting F : U −→ S 7 s.t. f = π ◦ F , where π : S 7 −→ CP3 is the Hopf fibration. The phrase ‘‘horizontal’’ means that Fz · F¯ = Fz¯ · F¯ = 0.

(3.1)

Since U is simply connected, two such lifting is uniquely determined up to a constant factor of absolute value 1. By using the fact that f is conformal and Lagrangian, we obtain Fz · Fz¯ = 0,

Fz · Fz = Fz¯ · Fz¯ = eu .

(3.2)

Notice that f (M ) ⊂ Q2 , we have F · F = 0,

(3.3)

which implies Fz · F = Fz¯ · F = 0.

(3.4)

From (3.1), (3.2) and (3.4), we know that F , Fz , Fz¯ and F¯ is a Hermitian moving frame along f . If we write F = (F , Fz , Fz¯ , F¯ ), then we obtain following equations

Fz = F A,

Fz¯ = F B,

(3.5)

where 0 1 A= 0 0



0 uz + φ e −u ψ

−α

−eu −φ¯ φ −β

0



e β , e−u γ¯ 0 −u ¯

0 0 B= 1 0



−eu −φ¯ φ −β

0 −eu ψ¯ uz¯ − φ¯

−γ

0



e α¯   e−u β¯ 0 −u

and

φ = e−u Fz z¯ · Fz¯ ,

ψ = Fzz · Fz¯ ,

α = Fz · Fz ,

β = Fz · Fz¯ ,

γ = Fz¯ · Fz¯ .

Proposition 3.1. The one-form Φ = φ dz and the cubic differential Ψ = ψ dz 3 is globally defined on M.

64

J. Wang, X. Xu / Journal of Geometry and Physics 97 (2015) 61–68

Proof. We only need to show that Φ , Ψ are independent of the choice of the holomorphic coordinate and the horizontal lifting. Let z = z (w) be a holomorphic coordinate transformation, i.e. ∂∂wz¯ =0. Then, we have

 ∂z   , eu˜ = eu  ∂w  2



ds2M = 2eu dzdz¯ = 2eu˜ dw dw, ¯ and Fz = Fw

∂w , ∂z

Fz¯ = Fw¯

∂w ¯ , ∂ z¯

   ∂w 2  . ∂z 

Fz z¯ = Fww¯ 

These identities imply that e−u Fz z¯ · Fz¯ dz = e−˜u Fww¯ · Fw¯ dw, ∂z by dz = ∂w dw . On the other hand, Φ is independent of the choice of the horizontal lifting by the fact that two horizontal lifting are different up to a constant factor of absolute 1. Similar proof for the cubic differential Ψ . This completes the proof. 

The compatibility condition Fz z¯ = Fz¯ z of the moving equations in (3.5) is equivalent to Az¯ − Bz = [A, B].

(3.6)

More explicitly, (3.6) is equivalent to the followings: uz z¯ + (φz¯ + φz¯ ) = −eu − |φ|2 + e−2u |ψ|2 + e−u (|α|2 − |β|2 ), ¯ ), −φz + e−u ψz¯ = −uz¯ φ¯ + e−u (αβ ¯ − βγ

−uz¯ β¯ + βz − αz¯ = φ α¯ + e−u ψ¯ γ¯ , e−u ψz¯ − φz = −uz φ + e−u (α β¯ − β γ¯ ), (φz¯ + φz¯ ) − uz z¯ = eu + |φ|2 − e−2u |ψ|2 + e−u (|β|2 − |γ |2 ), uz β¯ − βz¯ + γz = e−u ψ α¯ + φ¯ γ¯ , −αz¯ + βz = α φ¯ + uz β + e−u ψγ , ¯ − uz¯ β + φγ , −βz¯ + γz = e−u ψα

(3.7)

|α|2 = |γ |2 . These equations are necessary and sufficient for the existence of the corresponding Lagrangian surfaces in Q2 . Now, we can state and prove our theorem: Theorem 3.2. Let f : M −→ Q2 be a Lagrangian conformal isometric immersion and Φ be the associated one-form defined above. Then (1) f is minimal if and only if Φ = 0; (2) f is H-minimal if and only if Φ is holomorphic. Proof. Choosing the moving frame e0 = F , e1 = e−u/2 Fz , e2 = e−u/2 Fz¯ , e3 = F¯ along f , and writing the metric ds2M = θ θ¯ with θ =

√

2eu/2 dz. From the first structure Eq. (2.10), we have

ρ = (uz¯ dz¯ − uz dz )/2.

(3.8)

By using (3.2), we obtain

√

√ ω01¯ =

2 2

θ,

ω02¯ =

2 2

θ¯ ,

which give that

√ a1 = b 2 =

2 2

,

a2 = b1 = 0.

(3.9)

Notice that ωk¯l = dek · e¯ l , we have

ω11¯ = [−e−u/2 Fz (uz dz + uz¯ dz¯ )/2 + e−u/2 (Fzz dz + Fz z¯ dz¯ )] · eu/2 Fz = (uz dz − uz¯ dz¯ )/2 + φ dz − φ¯ dz¯ .

(3.10)

Similarly, we also have

ω12¯ = e−u ψ dz + φ dz¯ ,

ω21¯ = −φ¯ dz − e−u ψ¯ dz¯ ,

ω22¯ = −(uz dz − uz¯ dz¯ )/2 + (φ dz − φ¯ dz¯ ).

(3.11)

J. Wang, X. Xu / Journal of Geometry and Physics 97 (2015) 61–68

65

From the definition of Dak in (2.13), together with (3.8)-(3.11), we have p1 = e−u/2 φ/2,

¯ 2, q1 = −e−u/2 φ/

p2 = e−3u/2 ψ/2,

q2 = e−u/2 φ/2.

(3.12)

So, according to (2.15), f is minimal if and only if qk = 0, i.e., φ = 0 or Φ = 0. Recalling the definition of hk in (2.17), using (3.9) and (3.12), we obtain

√

√ h1 =

¯ 2, 2ieu/2 (φ − φ)/

h2 =

¯ 2. 2ieu/2 (φ + φ)/

(3.13)

Then, the covariant differential of Dhk are given by Dh1 = pθ + qθ¯ ,

Dh1 = r θ + sθ¯

where

¯ z ]/2, p = ie−u [−φ uz + (φ − φ) ¯ z ]/2, r = ie [φ uz − (φ + φ)

¯ z¯ ]/2, q = ie−u [φ¯ uz¯ + (φ − φ)

(3.14)

¯ z¯ ]/2. s = ie [φ¯ uz¯ − (φ + φ)

−u

−u

(3.15)

So, p + q + r − s = 0 if and only if

φz¯ − φz¯ = 0.

(3.16)

On the other hand, from the first, fifth and ninth equations in the compatibility condition (2.17), we have

φz¯ + φz¯ = 0.

(3.17)

Therefore, we have proved that f is H-minimal if and only if φz¯ = 0, i.e., Φ is holomorphic. This completes the proof.



Remark. According to the geometric meaning of qk , we call Φ is the holomorphic mean curvature form of f . The cubic differential Ψ is called the Hop differential of f . An immediately corollary is Corollary 3.3. Any H-minimal Lagrangian two-sphere in Q2 is also minimal. Proof. By the known fact that there is no non-trivial holomorphic one-form on S 2 . This completes the proof.



4. A family of minimal Lagrangian tori in Q2 In this section, we construct a family of minimal Lagrangian tori in Q2 by solving ODE-system. Suppose that M = T 2 = C/Γ is a torus with metric ds2 = 2eu dzdz¯ , f : C/Γ −→ Q2 is minimal and equivalent under the S 1 -action

ρ : S 1 −→ SO(4),

eit → diag



cos σ t − sin σ t

sin σ t cos σ t

  cos τ t , − sin τ t

sin τ t cos τ t



.

More explicitly, f (x, y + t ) = f (x, y)ρ(eit ) with z = x + iy is the complex coordinate on T 2 = C/Γ . Then, it is easy to check that u, ψ , α , β and γ are only dependent on the variable x. We will denote the derivative with respect to x by ′ in the sequel. To find a solution of (3.7), we further assume that α = γ = 0. Notice that φ = 0, then (3.7) becomes 1 ′′ u + eu − e−2u |ψ|2 + e−u |β|2 = 0, 4 ψ ′ = 0,

(4.1)

u β − β = 0. ′

′

The second equation in (4.1) implies that ψ = c ∈ C is constant. The third equation gives that β = c1 eu , where c1 ∈ C is a constant. Substituting ψ = c and β = c1 eu into the first equation in (4.1), we obtain 1 ′′ u + (1 + |c1 |2 )eu − |c |2 e−2u = 0. 4

(4.2)

Multiplication by u′ on both sides of (4.2) and integration yields to 1 8

1

(u′ )2 + (1 + |c1 |2 )eu + |c |2 e−2u = a

where a is a constant.

2

(4.3)

66

J. Wang, X. Xu / Journal of Geometry and Physics 97 (2015) 61–68

We wish to find a solution u = u(x) with the initial condition u′ (0) = 0. So, it follows from (4.3) that

|c |2 2a21

+ (1 + |c1 |2 )a1 = a,

(4.4)

where a1 = eu(0) . Making change the variable w = eu , from (4.3), we obtain

(w ′ )2 + 8(1 + |c1 |2 )w 3 − 8aw2 + 4|c |2 = 0. Notice that a1 = e

u(0)

(4.5)

= w(0) > 0 is a solution of 8(1 + |c1 | )w − 8aw + 4|c | = 0, we have 2

3

2

2

(w ′ )2 + 8(1 + |c1 |2 )(w − a1 )(w − a2 )(w − a3 ) = 0,

(4.6)

where

a2 =

a3 =

a − a1 (1 + |c1 |2 ) +



a1 (1 + |c1 |2 − a) +



2|c |2 (1+|c1 |2 ) a1

[a − a1 (1 + |c1 |2 )]2 +

2(1 + |c1 |2 ) 2|c |2 (1+|c1 |2 ) a1

[a − a1 (1 + |c1 |2 )]2 +

2(1 + |c1 |2 )

> 0, > 0.

So, in terms of the Jacobi elliptic function (see page 209-211 in [20] for details), the solution of (4.6) is given by

w = a1 − (1 − q2 sn2 (rx, p)),

(4.7)

where p2 =

a1 − a2 a1 + a3

,

q2 =

a1 − a2 a1

,

r =



2(a1 + a3 ).

By the properties of Jacobi elliptic functions, there exists a positive real number T = K /r s.t. u(x + 2T ) = u(x), u(T ) = a2 and u′ (T ) = 0, where π/2

 K = 0

ds



1 − p2 sin2 s

is the elliptic integral of the first kind. By using the equations in (3.5), we obtain Fxx = −2eu F +

1 ′ u Fx + ice−u Fy − 2c1 eu F¯ ,

2

1 ′ u Fy , 2

Fxy = −ice−u Fx − Fyy = −2eu F −

(4.8)

1 ′ u Fx − ice−u Fy − 2c1 eu F¯ . 2

On the other hand, since f is equivalent under the S 1 -action, we can write F (x, y) as





F (x, y) = F1 (x), F2 (x), F3 (x), F4 (x) ρ(eiy ). Then, from (4.8), we obtain

σ

σ F1 = −ice−u F2′ − τ F3 = −ice−u F4′ −

u′ F1 ,

σ F2 = ice−u F1′ −

u′ F3 ,

τ F4 = ice−u F3′ −

2

τ

2

σ 2

τ

2

u′ F 2 ,

u′ F4 .

(4.9) (4.10)

Notice that f (T 2 ) ⊂ Q2 , we further assume that F2 = iF1 and F4 = iF3 . Then (4.9) and (4.10) are equivalent to

σ (σ − ce−u )F1′ = − u′ F1 , 2

τ (τ − ce−u )F3′ = − u′ F3 .

Set F1 = r1 (x)eiv1 (x) ,

F3 = r3 (x)eiv3 (x) ,

2

(4.11)

J. Wang, X. Xu / Journal of Geometry and Physics 97 (2015) 61–68

67

then equations in (4.11) are equivalent to

(log r1 )′ = −

σ u′ (σ − Rec e−u ) , 2[(σ − Rec e−u )2 + (Imc )2 e−2u ]

v1′ = −

σ Imc u′ e−u , 2[(σ − Rec e−u )2 + (Imc )2 e−2u ]

(4.12)

(log r3 )′ = −

τ u′ (τ − Rec e−u ) , 2[(τ − Rec e−u )2 + (Imc )2 e−2u ]

v3′ = −

τ Imc u′ e−u . 2[(τ − Rec e−u )2 + (Imc )2 e−2u ]

(4.13)

By integrating (4.12) and (4.13), we have F3 (x) = l3 r˜3 (x)eiv˜ 3 (x) ,

F1 (x) = l1 r˜1 (x)eiv˜ 1 (x) ,

(4.14)

where



r˜1 (x) = exp −

v˜ 1 (x) =

1 2

arctan



1 2

4



log e2u − 2



r˜3 (x) = exp −

v˜ 3 (x) =

1

1 4

σ

eu +

− σ Rec σ Imc

2 −u

|c | e 

log e2u − 2

 arctan

Rec

Rec

τ







|c |2 τ2



−

Rec



 arctan

2Imc

σ eu − Rec





|c |2 e−u − σ Rec σ Imc

 ,



|c |2 e−u − τ Rec τ Imc

,

+ arctan

Imc

,

eu +

|c |2 e−u − τ Rec τ Imc

|c |2 σ2

−

Rec 2Imc



 arctan

τ eu − Rec



Imc

+ arctan



,

and the constant numbers l1 and l3 satisfying 2(|l1 |2 r12 (0) + |l3 |2 r32 (0)) = 1. Here, we need to point out that the imaginary of c is nonzero. The explicit expression of r˜i (x) and v˜ i (x) are much more simple if Imc = 0, we omit the details. Therefore, we conclude that





F (x, y) = l1 r˜1 ei(˜v1 (x)−σ y) , il1 r˜1 ei(˜v1 (x)−σ y) , l3 r˜3 ei(˜v3 (x)−τ y) , il3 r˜3 ei(˜v3 (x)−τ y) .

(4.15)

Now, we have Theorem 4.1. Assume F (x, y) defined by (4.15), then f = π ◦ F : R2 −→ Q2 ⊂ CP3 is a minimal Lagrangian immersion, where π : S 7 −→ CP3 is the Hopf fibration. Notice that r˜1 (x), v˜ 1 (x), r˜3 (x) and v˜ 3 (x) are expressed explicitly in terms of the periodic function u(x), and eiσ y , eiτ y are periodic functions with period 2σπ , 2τπ respectively. So, we obtain Theorem 4.2. The immersion f given in Theorem 4.1 is doubly periodic if and only if στ ∈ Q. Acknowledgments This work was supported by NSFC (Grant No.11471299, 11271343, 11226079). The authors would like to thank professors C.K. Peng and X.X. Jiao for their constant encouragements, and the second author would like to thank professor H. Ma for her valuable suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

Y.G. Oh, Volume minimization of Lagrangian submanifolds under Hamiltonian deformations, Math.Z. 212 (1993) 175–192. I. Castro, H. Li, F. Urbano, Hamiltonian-minimal Lagrangian submainifolds in complex space forms, Pacific J. Math. 227 (2006) 43–63. I. Castro, F. Urbano, New examples of minimal Lagrangian tori in the complex projective plan, Manuscriota Mathematicae 85 (1994) 265–281. I. Castro, F. Urbano, Examples of unstable Hamiltonian-minimal Lagrangian tori in C2 , Compositio Mathematicae 111 (1998) 1–14. B.Y. Cheng, O.J. Garay, Classification of Hamiltonian-stationary Lagrangian submanifolds of constant curvature in CP3 with positive nullity, Nonlinear Anal. 71 (2009) 2640–2659. R. Chiang, New Lagrangian submanifolds of CPn , Int.Math.Res.Not. 45 (2004) 2437–2441. Y.X. Dong, Hamiltonian-minimal Lagrangian submanifolds in Kaehler manifolds with symmetries, Nonlinear Anal. 67 (2007) 865–882. F. Helen, P. Rommon, Hamiltonian stationary Lagrangian surfaces in C2 , Commun. Anal. Geom. 10 (2002) 79–126. F. Helen, P. Rommon, Hamiltonian stationary tori in CP2 , Proc. Lond. Math. Soc. 90 (2005) 472–496. H. Ma, Hamiltonian stationary surfaces in CP2 , Ann. Global Anal. Geom. 27 (2005) 1–16. H. Ma, Y.J. Ma, Totally real minimal tori in CP2 , Math.Z. 249 (2005) 241–267. H. Ma, M. Schmies, Examples of Hamiltonian sationary Lagrangian tori in CP2 , Geom. Dedicata 118 (2006) 173–183. A.E. Mironov, New examples of Hamiltonian-minimal and minimal Lagrangian manifolds in Cn and CPn , Sobornik Math. 195 (2004) 89–102. A.E. Mironov, On the Hamiltonian-minimal Lagrangian tori in the CP2 , Sobornik Math. J. 44 (2003) 1039–1042.

68

J. Wang, X. Xu / Journal of Geometry and Physics 97 (2015) 61–68

[15] A.E. Mironov, The Veselov-Novikov hierachy of equations, and integrable deformations of minimal Lagrangian tori in CP2 , Sib. Elektron. Mat. Izv. 1 (2004) 38–46. [16] A.E. Mironov, On a family of conformal flat minimal Lagrangian tori in CP3 , Math. Notes 81 (2007) 329–337. [17] A.E. Mironov, D.F. Zuo, On a family of conformal flat Hamiltonian-minimal tori in CP3 , International Matematics Research Notices, Vol. 2008, Article ID rnn078, 13 pages. http://dx.doi.org/10.1093/imrn/rnn078. [18] K. Yang, Complete and Compact Minimal Surface, Kluwer Academic, Dordrecht, 1989. [19] Q. Chen, S. Hu, X.W. Xu, Construction of Lagrangian submanifolds in CPn , Pacific J. Math. 258 (1) (2012) 31–49. [20] H.T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover Publications, New York, 1962.