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Lagrangian submanifolds in complex space forms attaining equality in a basic inequality
J. Math. Anal. Appl. 387 (2012) 139–152
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Lagrangian submanifolds in complex space forms attaining equality in a basic inequality ✩ Bang-Yen Chen a,∗ , Franki Dillen b , Luc Vrancken b,c a b c
Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA Katholieke Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200 B, Box 2400, BE-3001 Leuven, Belgium LAMAV, ISTV2, Université de Valenciennes, Campus du Mont Houy, 59313 Valenciennes Cedex 9, France
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 21 May 2011 Available online 31 August 2011 Submitted by H.R. Parks
Lagrangian submanifolds appear naturally in the context of classical mechanics. Moreover, they play some important roles in supersymmetric field theories as well as in string theory. Recently, it was proved in Chen and Dillen (2011) [11] that for any Lagrangian sub˜ n (4c ), n 3, of constant holomorphic sectional manifold M of a complex space form M curvature 4c we have
Keywords: Lagrangian submanifold Optimal inequalities δ -Invariants
δ(n − 1)
n − 1 2 nH + 4c , 4
where H 2 is the squared mean curvature and δ(n − 1) is a δ -invariant of M (cf. Chen, 2000, 2011 [7,10]). In this paper, we completely classify non-minimal Lagrangian submanifolds of ˜ n (4c ), c = 0, 1, −1, which satisfy the equality case of the inequality complex space forms M identically. © 2011 Elsevier Inc. All rights reserved.
1. Introduction
˜ n be a complex n-dimensional Kähler manifold with the complex structure J and the metric g. The Kähler 2-form ω Let M ˜ n of a Riemannian n-manifold M into M ˜ n is called Lais defined by ω(·,·) = g ( J ·,·). An isometric immersion ψ : M → M ∗ grangian if ψ ω = 0. Such submanifolds appear naturally in the context of classical mechanics and mathematical physics. For instance the systems of partial differential equations of Hamilton–Jacobi type lead to the study of Lagrangian submanifolds and foliations in the cotangent bundle. Lagrangian submanifolds also play some important roles in supersymmetric field theories and in string theory. Denote by K (π ) the sectional curvature of the Riemannian n-manifold M associated with a plane section π ⊂ T p M, p ∈ M. For any orthonormal basis e 1 , . . . , en of the tangent space T p M, the scalar curvature τ at p is defined to be
τ ( p) =
K (e i ∧ e j ).
i< j
✩
*
This research was supported by a Research Grant of the Research Foundation–Flanders (FWO). Corresponding author. E-mail addresses: [email protected] (B.-Y. Chen), [email protected] (F. Dillen), [email protected] (L. Vrancken).
0022-247X/$ – see front matter doi:10.1016/j.jmaa.2011.08.066
©
2011 Elsevier Inc. All rights reserved.
(1.1)
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Let L be a subspace of T p M of dimension r 2 and {e 1 , . . . , er } an orthonormal basis of L. The scalar curvature the r-plane section L is defined by
τ (L) =
K (e α ∧ e β ),
1 α, β r.
τ ( L ) of (1.2)
α <β
For a given integer r with 2 r < n, the δ -invariant δ(r ) of a Riemannian n-manifold M is defined by
δ(r )( p ) = τ ( p ) − inf τ ( L ) ,
(1.3)
where L run over all r-dimensional linear subspaces of T p M. (The invariant δ(2) was first introduced in [4]. For general δ -invariants, see [7,9,10] for details.) Recently, the following result was proved in [11] (for δ(2), see [13]).
˜ n (4c ) of constant holomorphic sectional curvature 4c. Theorem A. Let M be a Lagrangian submanifold in a complex space form M Then for any integer r with 2 r < n we have
δ(r )
n2 {rn − r 2 + 2n − 2} 2{rn − r 2 + 2n + 3r + 4}
H2 +
1 n(n − 1) − r (r − 1) c , 2
(1.4)
where H 2 is the squared mean curvature of M. It was also proved in [11] that
˜ n (4c ). If M satisfies the equality case Theorem B. Let M be an n-dimensional Lagrangian submanifold of a complex space form M ˜ n (4c ). of (1.4) for some r n − 2, then M is a minimal submanifold of M When r = n − 1, we have the following result from [11].
˜ n (4c ). Then Theorem C. Let M be a Lagrangian submanifold of a complex space form M
δ(n − 1)
n − 1 2 nH + 4c . 4
(1.5)
The equality sign holds at a point p ∈ M if and only if there is an orthonormal basis {e 1 , . . . , en } at p such that with respect to this basis the second fundamental form h takes the following form
h(e 1 , e 1 ) = λ J e 1 , h(e α , e β ) =
h(e 1 , e α ) =
n γ
hα β J eγ +
γ =2
λ n+1
δα β λ J e1 , n+1
J eα , n γ
hαα = 0,
(1.6)
α =2
for α , β = 2, . . . , n and λ = h111 . For a Lagrangian submanifold M satisfying the equality case of (1.5) identically, we call an orthonormal frame {e 1 , . . . , en } an adapted frame if it satisfies (1.6). Minimal Lagrangian submanifolds of complex space forms satisfying the equality case of (1.4) (in particular, of (1.5)) were classified in [8]. On the other hand, by applying the notion of complex extensors introduced in [6], the first two authors constructed some explicit examples of non-minimal Lagrangian submanifolds in the complex Euclidean n-plane Cn which satisfy the equality case of (1.5). ˜ n (4c ) (c = In this paper, we completely classify all non-minimal Lagrangian submanifolds of complex space forms M 0, 1, −1) which satisfy the equality case of the inequality (1.5) identically. 2. Preliminaries 2.1. Basic formulas
˜ n (4c ) be a complete, simply-connected, Kähler n-manifold with constant holomorphic sectional curvature 4c and M Let M ˜ n (4c ). We denote the Levi-Civita connections of M and M ˜ n (4c ) by ∇ and ∇˜ , respectively. a Lagrangian submanifold of M
B.-Y. Chen et al. / J. Math. Anal. Appl. 387 (2012) 139–152
141
The formulas of Gauss and Weingarten are given respectively by
∇˜ X Y = ∇ X Y + h( X , Y ), ∇˜ X ξ = − A ξ X + D X ξ,
(2.1) (2.2)
for tangent vector fields X and Y and normal vector fields ξ , where D is the normal connection. The second fundamental form h is related to A ξ by
h( X , Y ), ξ = A ξ X , Y .
The mean curvature vector H of M is defined by
H=
1 n
trace h.
For Lagrangian submanifolds, we have (cf. [10,12])
D X J Y = J ∇X Y ,
(2.3)
A J X Y = − J h( X , Y ) = A J Y X .
(2.4)
The above formulas immediately imply that h( X , Y ), J Z is totally symmetric. If we denote the curvature tensors of ∇ and D by R and R D , respectively, then the equations of Gauss and Codazzi are given by
R ( X , Y ) Z , W = A h(Y , Z ) X , W − A h( X , Z ) Y , W + c X , W Y , Z − X , Z Y , W ,
(∇ h)( X , Y , Z ) = (∇ h)(Y , X , Z ), where X , Y , Z , W (respectively,
(2.5) (2.6)
η and ξ ) are vector fields tangent (respectively, normal) to M; and ∇ h is defined by
(∇ h)( X , Y , Z ) = D X h(Y , Z ) − h(∇ X Y , Z ) − h(Y , ∇ X Z ).
(2.7)
For an orthonormal basis {e 1 , . . . , en } of T p M at a point p ∈ M, we put
A h BC = h(e B , e C ), J e A ,
A , B , C = 1, . . . , n .
It follows from (2.4) that A h BC = h BAC = h CA B .
(2.8)
2.2. Horizontal lift of Lagrangian submanifolds The following link between Legendre submanifolds and Lagrangian submanifolds is due to [14]. Case (i): CPn (4). Consider Hopf’s fibration π : S 2n+1 (1) → CPn (4). Then π is a Riemannian submersion. Consider S 2n+1 (1) as a Sasakian manifold with the canonical Sasakian structure with contact form η induced from the complex structure of Cn+1 . An n-dimensional submanifold N of S 2n+1 (1) is called a Legendre submanifold if η( X ) = 0 for every vector tangent to N. Given z ∈ S 2n+1 (1), the horizontal space at z is the orthogonal complement of iz with respect to the metric on S 2n+1 (1) induced from the metric on Cn+1 . Let ψ : N → CPn (4) be a Lagrangian immersion. Then there exists a covering map τ : Nˆ → N and a horizontal immersion ψˆ : Nˆ → S 2n+1 (1) such that ψ ◦ τ = π ◦ ψˆ . Thus each Lagrangian immersion can be lifted locally (or globally if N is simply-connected) to a Legendre immersion of the same Riemannian manifold. ˆ → S 2n+1 (1) is a Legendre immersion, then ψ = π ◦ φ : N → CPn (4) is again a Lagrangian immersion. Conversely, if φ : N Under this correspondence the second fundamental forms hψ and hφ of ψ and φ satisfy π∗ hφ = hψ . Moreover, hφ is horizontal with respect to π . +1 denote the complex pseudo-Euclidean (n + 1)-space endowed with the pseudo-Euclidean Case (ii): CHn (−4). Let C2n 1 metric
g 0 = −dz1 d z¯ 1 +
n +1
dz j d z¯ j .
j =2
Consider the anti-de Sitter spacetime
+1 H 12n+1 (−1) = z ∈ C2n : b1,n+1 ( z, z) = −1 1
with the canonical Sasakian structure (cf. [10, p. 245]). ¯ = 1}, where , is the Hermitian inner product on Cn+1 whose real part Put T z = {u ∈ Cn+1 : u , z = 0}, H 11 = {λ ∈ C: λλ 1
is g 0 . Then there is an H 11 -action on H 12n+1 (−1), z → λ z and at each point z ∈ H 12n+1 (−1), the vector ξ = −iz is tangent
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to the flow of the action. Since the metric g 0 is Hermitian, we have ξ, ξ = −1. The quotient space H 12n+1 (−1)/ ∼, under the identification induced from the action, is the complex hyperbolic space CHn (−4) with constant holomorphic sectional curvature −4, with the complex structure J induced from the complex structure J on Cn1+1 via π : H 12n+1 (−1) → CHn (4c ).
ˆ → N and Just like case (i), if ψ : N → CHn (−4) is a Lagrangian immersion, then there is an isometric covering map τ : N ˆ → H 2n+1 (−1) such that ψ ◦ τ = π ◦ φ . Thus every Lagrangian immersion can be lifted locally a Legendre immersion φ : N 1 (or globally if N is simply-connected) to a Legendre immersion. 2n + 1 ˆ →H (−1) is a Legendre immersion, then ψ = π ◦ φ : N → CHn (−4) is a Lagrangian immersion. Conversely, if φ : N 1 Similarly, under this correspondence the second fundamental forms hφ and hψ are related by π∗ hφ = hψ . Also, hφ is horizontal with respect to π . ˆ the Levi-Civita connections Assume M is a submanifold of S 2n+1 (1) ⊂ Cn+1 (or in H 12n+1 (−1) ⊂ Cn1+1 ). Denote by ∇ of Cn+1 (or of Cn1+1 ) and denote by L the corresponding immersion of M in Cn+1 (or of Cn1+1 ).
Let h be the second fundamental form of M in S 2n+1 (1) (or in H 12n+1 (−1)). Since S 2n+1 (1) and H 12n+1 (−1) are totally
umbilical with one as its mean curvature in Cn+1 and in Cn1+1 , respectively, we have
∇ˆ X Y = ∇ X Y + h( X , Y ) − ε L , where
(2.9) n+1
ε = 1 whenever the ambient space is C
; and
ε = −1 whenever the ambient space is
Cn1+1 .
3. Some lemmas We need the following lemmas for the proof of the main results.
˜ n (4c ) satisfying the equality case of (1.5) identically and let Lemma 3.1. Let M be a non-minimal Lagrangian submanifold of M {e 1 , . . . , en } be an adapted frame satisfying (1.6). Then we have (a) ∇e1 e 1 = 0; (b) e α λ = 0; (c) ∇eα e 1 = μe α with e 1 λ = (n − 1)λμ; for α = 2, . . . , n. Proof. Under the hypothesis, we have λ = 0. Moreover, we may choose e 2 such that
∇e1 e 1 = b1 e 2 .
(3.1)
By putting V = Span{e 3 , . . . , en }, it follows from (1.6), (2.3), (2.7), (3.1) and
(∇ h)(e 1 , e α , e 1 ) = (∇ h)(e α , e 1 , e 1 ) that
e1 λ n+1 for
J eα +
λ J ∇e1 e α − b1 h(e 2 , e α ) − h(e 1 , ∇e1 e α ) = e α J e 1 + λ J ∇eα e 1 − 2h(∇eα e 1 , e 1 ) n+1
α = 2, . . . , n. Thus, by comparing the J e 1 -component of (3.2), we find
n−1 e2 λ = b1 λ, n+1 V λ = 0, V ∈ V .
(3.2)
(3.3) (3.4)
Similarly, by comparing the J V -component of (3.2), we have
(n + 1)b1 J h(e 2 , e 2 )⊥ , n−1 λ λ (n + 1)b1 1 e1 λ ∇V e1 = V + J h(e 2 , V ) , n−1 λ λ ∇e2 e 1 =
1
e1 λ
e2 +
where h( X , Y )⊥ denotes the component of h( X , Y ) orthogonal to J e 1 , i.e.
h( X , Y )⊥ = h( X , Y ) − h( X , Y ), J e 1 J e 1 . After applying (3.1), (3.3)–(3.6), we find from
(∇ h)(e 2 , e j , e 1 ), J e j = (∇ h)(e j , e 2 , e 1 ), J e j
(3.5) (3.6)
B.-Y. Chen et al. / J. Math. Anal. Appl. 387 (2012) 139–152
143
that
0 = λ2 b 1 +
(n + 1)3 b1 h(e j , e j ), h(e 2 , e 2 )⊥ − h(e 2 , e j ), h(e 2 , e j ) (n − 1)2
(3.7)
for j = 3, . . . , n. Also, we obtain from
(∇ h)(e 1 , e j , e j ), J e 2 = (∇ h)(e j , e 1 , e j ), J e 2 ,
j = 3, . . . , n ,
that
0 = D e1 h(e j , e j ), J e 2 − 2
+
e1 λ
(n − 1)λ
n
∇e1 e j , e α h(e j , e α ), J e 2
α =2
h(e j , e j ), J e 2 −
(n + 1)b1 h(e j , e 2 ), h(e j , e 2 ) . (n − 2)λ
(3.8)
Furthermore, we derive from (∇ h)(e 1 , e 2 , e 2 ), J e 2 = (∇ h)(e 2 , e 1 , e 2 ), J e 2 that
0 = D e1 h(e 2 , e 2 ), J e 2 − 2
+
e1 λ
(n − 1)λ
n
α =2
(n + 3)b1 λ ∇e1 e 2 , e α h(e 2 , e α ), J e 2 + (n + 1)2
h(e 2 , e 2 ), J e 2 −
(n + 1)b1 h(e 2 , e 2 )⊥ , h(e 2 , e 2 )⊥ . (n − 1)λ
(3.9)
After taking the sum of (3.8) over j = 3, . . . , n and adding it with (3.9), we find
0=
n (n2 + n + 2)b1 λ (n + 1)b1 − h(e 2 , e α )⊥ , h(e 2 , e α )⊥ . (n − 1)λ (n + 1)2
(3.10)
α =2
On the other hand, by taking the sum of (3.7) over j = 3, . . . , n, we derive that
0=
n (n2 − 3n + 2)b1 λ (n + 1)b1 − h(e 2 , e α )⊥ , h(e 2 , e α )⊥ . 2 (n − 1)λ (n + 1)
(3.11)
α =2
Therefore, after combining (3.10) and (3.11), we obtain b1 = 0. Consequently, we have Lemma 3.1(a) by (3.1). Lemma 3.1(b) follows from (3.3), (3.4) and Lemma 3.1(a). Finally, Lemma 3.1(c) follows from (3.5), (3.6) and b1 = 0. 2 A foliation L on a Riemannian manifold M is called totally umbilical if every leaf of L is a totally umbilical submanifold of M. If, in addition, the mean curvature vector of every leaf is parallel in the normal bundle, then L is called a spheric foliation. If every leaf of L is a totally geodesic submanifold of M, then L is called a totally geodesic foliation. Lemma 3.2. Under the hypothesis and notations of Lemma 3.1, we have (i) the distribution D1 = Span{e 1 } is a totally geodesic foliation; (ii) the distribution D2 = Span{e 2 , . . . , en } is a spherical foliation; (iii) M is locally the warped product I × f N of an integral curve I of e 1 and a leaf of D2 , where f is the warping function. Proof. Since D1 is a one-dimensional distribution, it is a foliation. It follows from Lemma 3.1(a) that D1 is a totally geodesic foliation. The integrability of the distribution D2 follows from Lemma 3.1(c). Moreover, we conclude from Lemma 3.1(b) and Lemma 3.1(c) that D2 is a spherical foliation. Statement (iii) follows from statements (i) and (ii) and a result of Hiepko (cf. e.g. [10, p. 90]). 2 Let us choose a local coordinate system {t , u 2 , . . . , un } on the Lagrangian submanifold M such that e 1 = ∂∂t and {u 2 , . . . , un } are local coordinates of N. Lemma 3.3. Under the hypothesis and notations of Lemma 3.1, we have
dλ dt dμ dt
= (n − 1)λμ, = −c − μ2 −
(3.12) n λ2
(n + 1)2
.
(3.13)
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B.-Y. Chen et al. / J. Math. Anal. Appl. 387 (2012) 139–152
Proof. From the equation of Gauss we have
for
R (e α , e 1 )e 1 , e α = c + h(e 1 , e 1 ), h(e α , e α ) − h(e 1 , e α ), h(e 1 , e α )
α = 2, . . . , n. On the other hand, it follows from (1.6) and Lemma 3.1 that R (e α , e 1 )e 1 , e α = −e 1 μ − μ2 , n λ2 h(e 1 , e 1 ), h(e α , e α ) − h(e 1 , e α ), h(e 1 , e α ) = . (n + 1)2
(3.14)
(3.15) (3.16)
Let us put e 1 = ∂∂t . Then, we obtain (3.12) from Lemma 3.1(c). Moreover, by combining (3.14), (3.15) and (3.16), we obtain (3.13). 2 In view of Lemma 3.1(c), we may put
∇eα e β = −μδα β e 1 +
n
γ
α , β = 2, . . . , n ,
Γα β e γ ,
(3.17)
γ =2 γ
for some functions Γα β . 4. Lagrangian submanifolds of Cn satisfying the equality The following theorem completely classifies non-minimal Lagrangian submanifold of the complex Euclidean n-space Cn satisfying the equality case of (1.5). Theorem 4.1. Let M be a non-minimal Lagrangian submanifold of the complex Euclidean n-space Cn . Then
1
δ(n − 1) n(n − 1) H 2 .
(4.1)
4
The equality sign holds identically if and only if up to dilations and rigid motions, M is defined by
L (λ, u 2 , . . . , un ) =
(n + 1)e −iϕ φ(u 2 , . . . , un ), (n + 1)μ + iλ
where
ϕ (λ) = −
n+1 n
n
csc−1 (n + 1)c λ 1−n ,
(4.2)
2
μ(λ) = c 2 λ 1−n − (n + 1)−2 λ2
(4.3)
for some positive real number c, and φ(u 2 , . . . , un ) is a minimal Legendre submanifold of S 2n−1 (1). Proof. Let M be a non-minimal Lagrangian submanifold of the complex Euclidean n-space Cn . Assume that M satisfies the equality case of (4.1) identically. Then there exists an adapted frame {e 1 , . . . , en } which satisfies (1.6). From Lemma 3.1 and Lemma 3.3 we have
∇eα e 1 = μe α , α = 2, . . . , n, dλ = (n − 1)λμ, dt dμ dt Let
= −μ2 −
n λ2
(n + 1)2
.
(4.4) (4.5) (4.6)
ϕ (t ) be a function satisfying dϕ dt
= −λ.
(4.7)
Consider the map
φ = e iϕ e 1 .
(4.8)
Then φ, φ = 1. It follows from (1.6) and (4.7) that
∇˜ e1 φ = 0,
(4.9)
B.-Y. Chen et al. / J. Math. Anal. Appl. 387 (2012) 139–152
145
˜ is the Levi-Civita connection of Cn . Thus, φ is independent of t. Let L be the Lagrangian immersion of M in Cn . where ∇ Then it follows from (4.8) that e1 =
∂L = e −iϕ φ(u 2 , . . . , un ), ∂t
(4.10)
where u 2 , . . . , un are local coordinates of N. For each α ∈ {2, . . . , n}, we get from (1.6) and Lemma 3.1(c) that
iϕ ˜ iϕ ˜ φ∗ (e α ) = ∇eα φ = e ∇eα e 1 = e μ +
iλ n+1
eα .
(4.11)
Thus, from (1.6), (3.17), and (4.11), we find
∇˜ eβ φ∗ (e α ) = e iϕ μ +
iλ n+1
∇˜ eβ e α
n γ γ Γα β + ihα β φ∗ (e γ ) − μ2 + = γ =2
=
λ2 δα β φ (n + 1)2
n γ γ Γα β + ihα β φ∗ (e γ ) − φ∗ (e α ), φ∗ (e β ) φ
(4.12)
γ =2
α , β = 2, . . . , n. Since M is a Lagrangian submanifold in Cn , (4.8) and (4.11) imply that iφ is perpendicular to each tangent space of M. Hence, φ is a horizontal immersion in the unit hypersphere S 2n−1 (1) ⊂ Cn . Moreover, it follows from (4.12) that the second fundamental form of φ is the original second fundamental form of M respect to the second factor N of the warped product I × f N. Hence, φ is a minimal horizontal immersion in S 2n−1 (1). Therefore, φ is a horizontal lift of a minimal Lagrangian immersion in CPn−1 (4). By direct computation, we find for
∇˜ e1 L − for
n+1
(n + 1)μ + iλ
e1
= ∇˜ eα L −
n+1
(n + 1)μ + iλ
e1
=0
α = 2, . . . , n. Thus, up to translations the Lagrangian immersion L is given by
(n + 1)e −iϕ φ(u 2 , . . . , un ), L= (n + 1)μ + iλ
where φ is a horizontal minimal immersion in S 2n−1 (1) and λ,
dλ dt
dϕ
= (n − 1)λμ,
dt
= −λ,
dμ dt
= −μ2 −
(4.13)
(4.14)
ϕ , μ satisfy
n λ2
(n + 1)2
.
(4.15)
From (4.15) we find
d
dt
λ
2 n −1
λ2 μ + (n + 1)2
2
= 0,
2
(4.16)
2
λ which implies that λ n−1 (μ2 + (n+ ) = c 2 for some positive number c. Thus 1)2
2
μ = ± c 2 λ 1−n −
λ2 , (n + 1)2
(4.17)
for some positive real number c. Replacing e 1 by −e 1 if necessary, we have
2
μ = c 2 λ 1−n −
λ2 . (n + 1)2
(4.18)
Now, it follows from (4.15) and (4.18) we get
dϕ dλ
=
1
2 (1 − n) c 2 λ 1−n −
. λ2 (n+1)2
(4.19)
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B.-Y. Chen et al. / J. Math. Anal. Appl. 387 (2012) 139–152
Thus we obtain
ϕ =b−
n+1 n
n
csc−1 (n + 1)c λ 1−n .
(4.20)
Therefore, after applying a suitable rotation of Cn , we get (4.2) and (4.3). The converse can be verified by straightforward computation. 2 A Lagrangian submanifold of Cn without totally geodesic points is called a Lagrangian H -umbilical submanifold if its second fundamental form takes the following simple form (cf. [5,6]):
h(¯e 1 , e¯ 1 ) = ϕ J e¯ 1 ,
h(¯e j , e¯ j ) = μ J e¯ 1 ,
h(¯e 1 , e¯ j ) = μ J e¯ j ,
h(¯e j , e¯ k ) = 0,
j > 1,
2 j = k n
(4.21)
for some functions ϕ , μ with respect to a suitable orthonormal local frame field {¯e 1 , . . . , e¯ n }. Such submanifolds are the simplest Lagrangian submanifolds next to the totally geodesic ones. Let G : N n−1 → En be an isometric immersion of a Riemannian (n − 1)-manifold into the Euclidean n-space En and let F : I → C∗ be a unit speed curve in C∗ = C − {0}. We may extend G : N n−1 → En to an immersion of I × N n−1 into Cn as
F ⊗ G : I × N n−1 → C ⊗ En = Cn ,
(4.22)
where ( F ⊗ G )(s, p ) = F (s) ⊗ G ( p ) for s ∈ I , p ∈ N extensor of G via F . The following result was proved in [6].
n−1
. This extension F ⊗ G of G via tensor product is called the complex
Proposition 4.1. Let ι : S n−1 → En be the inclusion of a hypersphere of Em centered at the origin. Then every complex extensor φ = F ⊗ ι of ι via a unit speed curve F : I → C∗ is a Lagrangian H -umbilical submanifold of Cn unless F is contained in a line through the origin. For F ⊗ ι, let us choose a unit vector field e¯ 1 tangent to the first factor and e¯ 2 , . . . , e¯ n tangent to the second factor of I × S n−1 . If we put F (s) = e iζ (s) and F (s) = r (s)e iθ(s) , then it follows from [6] that the second fundamental form of F ⊗ ι satisfies (6.1) with
ϕ = ζ (s) = κ ,
μ=
F , iF = θ (s), F , F
(4.23)
where κ is the curvature function of F . By applying (4.21), (4.23) and Theorem C, we conclude that if the unit speed curve F : I → C satisfies
κ (s) = (n + 1)θ (s) = 0, s ∈ I , then the complex extensor F ⊗ ι : I × S case of (1.5). Remark 4.1. The curve
γ (λ) =
(4.24) n−1
→ C is a non-minimal Lagrangian submanifold of C which verifies the equality n
n
(n+1)e iϕ (n+1)μ+iλ in C defined by (4.2) and (4.3) satisfies condition (4.24).
Remark 4.2. When φ(u 2 , . . . , un ) in (4.2) is a totally geodesic Legendre submanifold of S n−1 (1), then (4.2) reduces to the complex extensor given above. 5. Lagrangian submanifolds of CPn (4) satisfying the equality Theorem 5.1. Let M be a non-minimal Lagrangian submanifold of the complex projective n-space CPn (4). Then
δ(n − 1)
n − 1 2 nH + 4 . 4
(5.1)
The equality sign holds identically if and only if M is congruent to the Lagrangian submanifold defined by the composition π ◦ L, where π : S 2n+1 (1) → CPn (4) is the Hopf fibration and
L (t , u 2 , . . . , un ) =
(i(n + 1)−1 λ − μ)e −niθ , , 1 + μ2 + (n + 1)−2 λ2 1 + μ2 + (n + 1)−2 λ2
where λ(t ), μ(t ) and θ(t ) satisfy
e −iθ φ
(5.2)
B.-Y. Chen et al. / J. Math. Anal. Appl. 387 (2012) 139–152
dθ dt dλ
=−
λ n+1
,
147
λ = 0,
= (n − 1)λμ,
dt dμ
= −1 − μ2 −
dt
n λ2
(5.3)
(n + 1)2
and φ is a minimal Legendre immersion in S 2n−1 (1). Proof. Let M be a non-minimal Lagrangian submanifold of the complex projective n-space CPn (4). Assume that M satisfies the equality case of (5.1) identically. Then there exists an adapted frame {e 1 , . . . , en } which satisfies (1.6). From Lemma 3.1 and Lemma 3.3 we have
∇eα e 1 = μe α , α = 2, . . . , n, dλ = (n − 1)λμ,
(5.4) (5.5)
dt
dμ
= −1 − μ2 −
dt
n λ2
(n + 1)2
(5.6)
.
Let L denote the immersion in Cn+1 associated with the horizontal lift of the Lagrangian immersion of M in CPn (4) via Hopf’s fibration π : S 2n+1 (1) → CPn (4) and let θ(t ) be a function on M satisfying
dθ dt
=−
λ . n+1
(5.7)
Consider the map:
ξ=
eniθ (e 1 − (μ + ni+λ1 ) L )
2
λ 1 + μ2 + (n+ 1 )2
(5.8)
.
˜ eα ξ = 0, α = 2, . . . , n, where ∇˜ is the Levi-Civita connection Then ξ, ξ = 1. Also, it follows from (1.6) and (5.4) that ∇ ˜ e1 ξ = 0. Hence, ξ is a constant unit vector of Cn+1 . Moreover, it follows from Lemma 3.1(a), (1.6), and (5.5)–(5.7) that ∇ in Cn+1 . Let us put e iθ ( L + (μ − ni+λ1 )e 1 ) φ= . λ2 1 + μ2 + (n+ 2 1)
(5.9)
˜ e1 φ = 0. Thus, φ is independent of t. Now, by combining Then, by applying Lemma 3.1(a), (1.6), and (5.5)–(5.7), we find ∇ (5.8) and (5.9), we get L=
e −iθ φ − e −niθ (μ − ni+λ1 )ξ
2
λ 1 + μ2 + (n+ 1 )2
(5.10)
.
Since φ is orthogonal to ξ , iξ , we obtain (5.2) after choosing ξ = (0, . . . , 0, 1) ∈ Cn+1 . It follows from (1.6), (5.4) and (5.9) that
∇ˆ eα φ = e iθ 1 + μ2 + for
λ2 eα (n + 1)2
(5.11)
α = 2, . . . , n. Thus, by applying (3.17) and (5.11) we find
λ2 ∇ˆ eβ ∇ˆ eα φ = e iθ 1 + μ2 + (∇ˆ eβ e α ) (n + 1)2
n γ λ2 γ iθ 2 Γα β + ihα β φ∗ (e γ ) − φ∗ (e α ), φ∗ (e β ) φ = e 1+μ + (n + 1)2 γ =2
(5.12)
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for α , β = 2, . . . , n. Hence, φ is a horizontal immersion in the unit hypersphere S 2n−1 (1) ⊂ Cn . Moreover, it follows from (5.12) that the second fundamental form of φ is a scalar multiple of the original second fundamental form of M restricted to the second factor of the warped product I × f N. Hence, φ is a minimal horizontal immersion in S 2n−1 (1). The converse can be verified by direct computation. 2 Remark 5.1. Theorem 5.1 extends the main result of [3] (see also [1,2]). 6. Lagrangian submanifolds of CH n (−4) satisfying the equality Finally, we completely classify non-minimal Lagrangian submanifolds of the complex hyperbolic n-space CHn (−4) satisfying the equality case of inequality (1.5). Theorem 6.1. Let M be a non-minimal Lagrangian submanifold of the complex hyperbolic n-space CHn (−4). Then
δ(n − 1)
n − 1 2 nH − 4 . 4
(6.1)
The equality sign holds identically if and only if M is congruent to the Lagrangian submanifold defined by the composition π ◦ L, where π : H 12n+1 (−1) → CHn (−4) is the Hopf fibration and L is one of the following immersions: (a) the immersion given by
L (t , u 2 , . . . , un ) =
(i(n + 1)−1 λ − μ)e −niθ , , 1 − μ2 − (n + 1)−2 λ2 1 − μ2 − (n + 1)−2 λ2 e −iθ φ
(6.2)
where λ(t ), μ(t ) and θ(t ) satisfy
dθ dt dλ dt dμ dt
=−
λ n+1
λ = 0,
,
1 > μ2 + (n + 1)−2 λ2 ,
= (n − 1)λμ, = 1 − μ2 −
n λ2
(6.3)
(n + 1)2
and φ is a minimal Legendre immersion in H 12n−1 (−1); (b) the immersion given by
L (t , u 2 , . . . , un ) =
e −iθ φ (i(n + 1)−1 λ − μ)e −niθ , , μ2 + (n + 1)−2 λ2 − 1 μ2 + (n + 1)−2 λ2 − 1
(6.4)
where λ(t ), μ(t ) and θ(t ) satisfy
dθ dt dλ dt dμ dt
=−
λ , n+1
λ = 0,
= (n − 1)λμ, = 1 − μ2 −
1 < μ2 + (n + 1)−2 λ2 , n λ2
(6.5)
(n + 1)2
and φ is a minimal Legendre immersion in S 12n−1 (1); (c) the immersion given by
L (t , u 2 , . . . , un ) =
2i 1 −1 e n−1 tan (tanh( 2 (n−1)t )) 1
cosh n−1 ((n − 1)t )
t
+
2
cosh 1−n (n − 1)t e
w+
i 2
i φ, φ + i, φ, w + φ, φ 2
2i tan−1 (tanh( 12 (n−1)t ))
dt (1, 0, . . . , 0, 1) ,
(6.6)
0
where φ(u 2 , . . . , un ) is a minimal Lagrangian immersion in Cn−1 and up to constants w (u 2 , . . . , un ) is the unique solution of the PDE system:
∂w ∂φ = i ,φ , ∂ uα ∂ uα
α = 2, . . . , n .
(6.7)
B.-Y. Chen et al. / J. Math. Anal. Appl. 387 (2012) 139–152
149
Proof. Let M be a non-minimal Lagrangian submanifold of the complex hyperbolic n-space CHn (−4). Assume that M satisfies the equality case of (6.1) identically. Then there exists an adapted frame {e 1 , . . . , en } which satisfies (1.6). From Lemma 3.1 and Lemma 3.3 we have
∇eα e 1 = μe α , α = 2, . . . , n, dλ = (n − 1)λμ,
(6.8) (6.9)
dt
dμ
= 1 − μ2 −
dt
n λ2
(n + 1)2
(6.10)
.
Let L be the immersion in Cn1+1 associated with the horizontal lift of the Lagrangian immersion in CHn (−4) via Hopf’s fibration
dθ dt
π : H 12n+1 (−1) → CHn (−4) and let θ(t ) be a function satisfying =−
λ n+1
(6.11)
.
It follows from (6.9) and (6.10) that
d
dt
1 − μ2 −
λ2 (n + 1)2
= −2μ 1 − μ2 −
λ2 . (n + 1)2
(6.12) 2
λ Since this first order differential equation for y (t ) = 1 − μ2 − (n+ has a unique solution for each given initial condition, 1)2 every solution is either vanishing identically or nowhere vanishing. Now, we divide the proof into three cases. 2
λ > 0. Consider the map: Case (i): 1 − μ2 − (n+ 1)2
η=
eniθ (e 1 − (μ + ni+λ1 ) L )
2
λ 1 − μ2 − (n+ 1 )2
(6.13)
.
˜ eα η = 0, α = 2, . . . , n, where ∇˜ is the Levi-Civita connection Then η, η = 1. Also, it follows from (1.6) and (6.8) that ∇ n+1 ˜ e1 ξ = 0 as well. Hence, η is a constant space-like of C1 . Moreover, it follows Lemma 3.1(a), (1.6), and (6.9)–(6.11) that ∇
unit vector in Cn1+1 . Let us put
e iθ ((μ − ni+λ1 )e 1 − L ) φ= . λ2 1 − μ2 − (n+ 1 )2
(6.14)
˜ e1 φ = 0. Thus, φ is independent of t. Now, by combining Then, by applying Lemma 3.1(a), (1.6), and (6.9)–(6.11), we find ∇ (6.13) and (6.14), we get L=
e −niθ (μ − ni+λ1 )η − e −iθ φ
2
λ 1 − μ2 − (n+ 1 )2
Since φ is orthogonal to
.
(6.15)
η, iη, we find (6.2) after choosing η = (0, . . . , 0, 1) ∈ Cn1+1 . Now, by applying similar arguments as
given in the proof of Theorem 5.1, we conclude that φ is a minimal Legendre immersion in H 12n−1 (−1). This gives case (i) of the theorem. 2
λ < 0. In this case, we consider the following maps: Case (ii): 1 − μ2 − (n+ 1)2
η=
eniθ (e 1 − (μ + ni+λ1 ) L )
2
μ2 + (n+λ 1)2 − 1
e iθ ((μ − ni+λ1 )e 1 − L ) φ= 2 μ2 + (n+λ 1)2 − 1
,
(6.16)
(6.17)
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instead. Then, after applying similar arguments as case (i), we may conclude that η is a constant unit time-like vector and φ is independent of t which is orthogonal to η , iη . Moreover, we may prove that φ is a minimal Legendre immersion in S 2n−1 (1). Thus, after choosing η = (1, 0, . . . , 0) we obtain case (ii) of the theorem. 2
λ = 0. By solving (6.9) and (6.10), we find Case (iii): 1 − μ2 − (n+ 1)2
λ = (n + 1) sech (1 − n)t ,
μ = tanh (1 − n)t .
(6.18)
It follows from ∇e1 e 1 = 0, (1.6) and (6.18) that the horizontal lift L of the Lagrangian immersion of M in CH (−4) satisfies n
L tt − i(n + 1) sech (1 − n)t L t − L = 0.
(6.19)
Solving this second order differential equation gives
2i 1 1 −1 L = cosh 1−n (n − 1)t e n−1 tan (tanh( 2 (n−1)t )) A + ρ (t )ζ ,
(6.20)
where A (u 2 , . . . , un ) and ζ (u 2 , . . . , un ) are Cn1+1 -valued functions and
t
ρ (t ) =
2 1 −1 cosh n−1 (n − 1)t e 2i tan (tanh( 2 (n−1)t )) dt .
(6.21)
0
By direct computation we have
t
ρ (t ) =
3−n
cosh n−1 (n − 1)t dt +
i 2
2
cosh n−1 (n − 1)t − 1 .
(6.22)
0
On the other hand, it follows from Lemma 3.1(c), (1.6) and (6.18) that
L tu α = tanh (1 − n)t + i sech (1 − n)t
L uα ,
By substituting (6.20) into (6.23), we find ζu α = 0, Since L , L = −1, it follows from (6.20) that
α = 2, . . . , n .
(6.23)
α = 2, . . . , n. Thus ζ is a constant vector in
Cn1+1 .
2 − cosh n−1 (n − 1)t = A , A + 2 A , ρ (t )ζ + ρ (t )ζ, ρ (t )ζ . Since A , A is independent of t and
(6.24)
ρ (0) = 0, (6.24) implies that
A , A = −1.
(6.25)
Hence, we get from (6.24) that
2 A , ρ (t )ζ +
2
ρ (t )ζ, ρ (t )ζ = 1 − cosh n−1 (n − 1)t .
(6.26)
Differentiating (6.26) with respect to t yields
A , ρ (t )ζ +
n −3
ρ (t )ζ, ρ (t )ζ = − sinh (n − 1)t sech n−1 (n − 1)t .
(6.27)
Thus, by taking t = 0, we find from (6.21) and (6.27) that
A , ζ = 0.
(6.28)
By applying (6.22), (6.26) and (6.28), we find
2
ρ (t )ζ, ρ (t )ζ = 1 − cosh n−1 (n − 1)t
1 + A , iζ .
(6.29)
Thus, after differentiating (6.29) with respect to t twice, we obtain
ρ (t )ζ, ρ (t )ζ + ρ (t )ζ, ρ (t )ζ = 2 − n − cosh (n − 1)t sech
2(n−2) n −1
(n − 1)t 1 + A , iζ .
(6.30)
Thus, at t = 0, (6.30) implies that
ζ, ζ = (1 − n) 1 + A , iζ .
(6.31)
B.-Y. Chen et al. / J. Math. Anal. Appl. 387 (2012) 139–152
151
On the other hand, it follows from (6.20) and (6.21) that
2i 1 1 −1 L t = cosh 1−n (n − 1)t e n−1 tan (tanh( 2 (n−1)t )) ρ (t )ζ + tanh (n − 1)t + i sech (n − 1)t A + ρ (t )ζ .
(6.32)
Since L is a Legendre immersion, we have L t , iL = 0. Thus we derive from (6.20) and (6.32) that
0=
ρ (t )ζ, i A + ρ (t )ζ + tanh (n − 1)t + i sech (n − 1)t
Since ρ (0) = 0, ρ (0) = 1 and
A + ρ (t )ζ , i A + ρ (t )ζ .
(6.33)
A , A = −1, (6.33) implies that
ζ, i A = 1.
(6.34)
By combining this with (6.31), we conclude that ζ, ζ = 0. Thus, ζ is a constant light-like vector. Without loss of generality, we may put
ζ = (1, 0, . . . , 0, 1).
(6.35)
Let us put
A = (a1 + ib1 , a2 + ib2 , . . . , an+1 + ibn+1 ).
(6.36)
Then it follows from (6.28) and (6.34)–(6.36) that an+1 = a1 and bn+1 = b1 − 1. Therefore
A = a1 + ib1 , a2 + ib2 , . . . , a1 + i(b1 − 1) .
(6.37)
Now, by using (6.25) and (6.37), we find
b1 =
1 2
φ, φ + 1.
(6.38)
By combining (6.37) and (6.38), we obtain
A=
w+
i 2
i φ, φ + i, φ, w + φ, φ ,
(6.39)
2
where w = a1 and φ = (a2 + ib2 , . . . , an + ibn ). Since L u α , iL = 0, it follows from (6.20) that
A u α , i A + A u α , iρ (t )ζ = 0,
α = 2, . . . , n .
(6.40)
Also, it follows from (6.35) and (6.39) that A u α , iρ (t )ζ = 0. On the other hand, (6.39) implies that
∂w ∂φ A uα , i A = + , iφ . ∂ uα ∂ uα
(6.41)
Thus the function w satisfies
∂w ∂φ = i ,φ . ∂ uα ∂ uα
(6.42)
Consequently, we obtain from (6.20), (6.35) and (6.37) that
i i L = H (t ) w + φ, φ + i, φ, w + φ, φ + ρ (t )(1, 0, . . . , 0, 1) , 2
2
(6.43)
where
H (t ) =
2i 1 −1 e n−1 tan (tanh( 2 (n−1)t )) 1
cosh n−1 ((n − 1)t )
(6.44)
,
and w (u 2 , . . . , un ) is a function satisfying (6.42). It follows from (6.43) that
L u α = H (t ) w u α + iφu α , φ , φu α , w u α + iφu α , φ . 2
(6.45)
Thus φu α , φu β = cosh n−1 ((n − 1)t ) L u α , L u β , which shows that φ is an immersion in Cn−1 . Also, we find from (6.44) and L u α , iL u β = 0 that φu α , iφu β = 0. Hence, φ is Lagrangian.
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Moreover, it follows from (6.43) that
i i e 1 = H (t ) w + φ, φ + i, φ, w + φ, φ + ρ (t )(1, 0, . . . , 0, 1) + H (t )ρ (t )(1, 0, . . . , 0, 1).
2
2
(6.46)
From (6.43) and (6.46) we find
(1, 0, . . . , 0, 1) = For
e1 H (t )ρ (t )
+
H (t ) H 2 (t )
L.
(6.47)
α , β = 2, . . . , n, we obtain from (1.6), (2.9), (3.17) and (6.18) that n γ γ Γα β + ihα β L uk + δα β L . ∇ˆ ea e β = δα β i sech (1 − n)t − tanh (1 − n)t e 1 +
(6.48)
γ =2
Therefore, (6.45), (6.47) and (6.48) give
φα β =
n γ γ Γα β + ihα β φuk .
(6.49)
γ =2
Hence, we may conclude from (1.6), (6.48) and (6.49) that φ is a minimal Lagrangian immersion in Cn−1 . Consequently, we obtain case (iii) of the theorem. The converse can be verified by direct computations. 2 References [1] J. Bolton, F. Dillen, J. Fastenakels, L. Vrancken, A best possible inequality for curvature-like tensor fields, Math. Inequal. Appl. 12 (2009) 663–681. [2] J. Bolton, C. Rodriguez Montealegre, L. Vrancken, Characterizing warped product Lagrangian immersions in complex projective space, Proc. Edinb. Math. Soc. 51 (2008) 1–14. [3] J. Bolton, L. Vrancken, Lagrangian submanifolds attaining equality in the improved Chen’s inequality, Bull. Belg. Math. Soc. Simon Stevin 14 (2007) 311–315. [4] B.-Y. Chen, Some pinching and classification theorems for minimal submanifolds, Arch. Math. 60 (1993) 568–578. [5] B.-Y. Chen, Interaction of Legendre curves and Lagrangian submanifolds, Israel J. Math. 99 (1997) 69–108. [6] B.-Y. Chen, Complex extensors and Lagrangian submanifolds in complex Euclidean spaces, Tohoku Math. J. 49 (1997) 277–297. [7] B.-Y. Chen, Some new obstruction to minimal and Lagrangian isometric immersions, Jpn. J. Math. 26 (2000) 105–127. [8] B.-Y. Chen, Ideal Lagrangian immersions in complex space forms, Math. Proc. Cambridge Philos. Soc. 128 (2000) 511–533. [9] B.-Y. Chen, Riemannian submanifolds, in: F. Dillen, L. Verstraelen (Eds.), Handbook of Differential Geometry, vol. I, North-Holland, Amsterdam, 2000, pp. 187–418. [10] B.-Y. Chen, Pseudo-Riemannian Geometry, δ -Invariants and Applications, World Scientific, Hackensack, NJ, 2011. [11] B.-Y. Chen, F. Dillen, Optimal general inequalities for Lagrangian submanifolds in complex space forms, J. Math. Anal. Appl. 379 (2011) 229–239. [12] B.-Y. Chen, K. Ogiue, On totally real submanifolds, Trans. Amer. Math. Soc. 193 (1974) 257–266. [13] T. Oprea, Chen’s inequality in the Lagrangian case, Colloq. Math. 108 (2007) 163–169. [14] H. Reckziegel, Horizontal lifts of isometric immersions into the bundle space of a pseudo-Riemannian submersion, Lecture Notes in Math. 1156 (1985) 264–279.
Contents lists available at SciVerse ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Lagrangian submanifolds in complex space forms attaining equality in a basic inequality ✩ Bang-Yen Chen a,∗ , Franki Dillen b , Luc Vrancken b,c a b c
Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA Katholieke Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200 B, Box 2400, BE-3001 Leuven, Belgium LAMAV, ISTV2, Université de Valenciennes, Campus du Mont Houy, 59313 Valenciennes Cedex 9, France
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 21 May 2011 Available online 31 August 2011 Submitted by H.R. Parks
Lagrangian submanifolds appear naturally in the context of classical mechanics. Moreover, they play some important roles in supersymmetric field theories as well as in string theory. Recently, it was proved in Chen and Dillen (2011) [11] that for any Lagrangian sub˜ n (4c ), n 3, of constant holomorphic sectional manifold M of a complex space form M curvature 4c we have
Keywords: Lagrangian submanifold Optimal inequalities δ -Invariants
δ(n − 1)
n − 1 2 nH + 4c , 4
where H 2 is the squared mean curvature and δ(n − 1) is a δ -invariant of M (cf. Chen, 2000, 2011 [7,10]). In this paper, we completely classify non-minimal Lagrangian submanifolds of ˜ n (4c ), c = 0, 1, −1, which satisfy the equality case of the inequality complex space forms M identically. © 2011 Elsevier Inc. All rights reserved.
1. Introduction
˜ n be a complex n-dimensional Kähler manifold with the complex structure J and the metric g. The Kähler 2-form ω Let M ˜ n of a Riemannian n-manifold M into M ˜ n is called Lais defined by ω(·,·) = g ( J ·,·). An isometric immersion ψ : M → M ∗ grangian if ψ ω = 0. Such submanifolds appear naturally in the context of classical mechanics and mathematical physics. For instance the systems of partial differential equations of Hamilton–Jacobi type lead to the study of Lagrangian submanifolds and foliations in the cotangent bundle. Lagrangian submanifolds also play some important roles in supersymmetric field theories and in string theory. Denote by K (π ) the sectional curvature of the Riemannian n-manifold M associated with a plane section π ⊂ T p M, p ∈ M. For any orthonormal basis e 1 , . . . , en of the tangent space T p M, the scalar curvature τ at p is defined to be
τ ( p) =
K (e i ∧ e j ).
i< j
✩
*
This research was supported by a Research Grant of the Research Foundation–Flanders (FWO). Corresponding author. E-mail addresses: [email protected] (B.-Y. Chen), [email protected] (F. Dillen), [email protected] (L. Vrancken).
0022-247X/$ – see front matter doi:10.1016/j.jmaa.2011.08.066
©
2011 Elsevier Inc. All rights reserved.
(1.1)
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B.-Y. Chen et al. / J. Math. Anal. Appl. 387 (2012) 139–152
Let L be a subspace of T p M of dimension r 2 and {e 1 , . . . , er } an orthonormal basis of L. The scalar curvature the r-plane section L is defined by
τ (L) =
K (e α ∧ e β ),
1 α, β r.
τ ( L ) of (1.2)
α <β
For a given integer r with 2 r < n, the δ -invariant δ(r ) of a Riemannian n-manifold M is defined by
δ(r )( p ) = τ ( p ) − inf τ ( L ) ,
(1.3)
where L run over all r-dimensional linear subspaces of T p M. (The invariant δ(2) was first introduced in [4]. For general δ -invariants, see [7,9,10] for details.) Recently, the following result was proved in [11] (for δ(2), see [13]).
˜ n (4c ) of constant holomorphic sectional curvature 4c. Theorem A. Let M be a Lagrangian submanifold in a complex space form M Then for any integer r with 2 r < n we have
δ(r )
n2 {rn − r 2 + 2n − 2} 2{rn − r 2 + 2n + 3r + 4}
H2 +
1 n(n − 1) − r (r − 1) c , 2
(1.4)
where H 2 is the squared mean curvature of M. It was also proved in [11] that
˜ n (4c ). If M satisfies the equality case Theorem B. Let M be an n-dimensional Lagrangian submanifold of a complex space form M ˜ n (4c ). of (1.4) for some r n − 2, then M is a minimal submanifold of M When r = n − 1, we have the following result from [11].
˜ n (4c ). Then Theorem C. Let M be a Lagrangian submanifold of a complex space form M
δ(n − 1)
n − 1 2 nH + 4c . 4
(1.5)
The equality sign holds at a point p ∈ M if and only if there is an orthonormal basis {e 1 , . . . , en } at p such that with respect to this basis the second fundamental form h takes the following form
h(e 1 , e 1 ) = λ J e 1 , h(e α , e β ) =
h(e 1 , e α ) =
n γ
hα β J eγ +
γ =2
λ n+1
δα β λ J e1 , n+1
J eα , n γ
hαα = 0,
(1.6)
α =2
for α , β = 2, . . . , n and λ = h111 . For a Lagrangian submanifold M satisfying the equality case of (1.5) identically, we call an orthonormal frame {e 1 , . . . , en } an adapted frame if it satisfies (1.6). Minimal Lagrangian submanifolds of complex space forms satisfying the equality case of (1.4) (in particular, of (1.5)) were classified in [8]. On the other hand, by applying the notion of complex extensors introduced in [6], the first two authors constructed some explicit examples of non-minimal Lagrangian submanifolds in the complex Euclidean n-plane Cn which satisfy the equality case of (1.5). ˜ n (4c ) (c = In this paper, we completely classify all non-minimal Lagrangian submanifolds of complex space forms M 0, 1, −1) which satisfy the equality case of the inequality (1.5) identically. 2. Preliminaries 2.1. Basic formulas
˜ n (4c ) be a complete, simply-connected, Kähler n-manifold with constant holomorphic sectional curvature 4c and M Let M ˜ n (4c ). We denote the Levi-Civita connections of M and M ˜ n (4c ) by ∇ and ∇˜ , respectively. a Lagrangian submanifold of M
B.-Y. Chen et al. / J. Math. Anal. Appl. 387 (2012) 139–152
141
The formulas of Gauss and Weingarten are given respectively by
∇˜ X Y = ∇ X Y + h( X , Y ), ∇˜ X ξ = − A ξ X + D X ξ,
(2.1) (2.2)
for tangent vector fields X and Y and normal vector fields ξ , where D is the normal connection. The second fundamental form h is related to A ξ by
h( X , Y ), ξ = A ξ X , Y .
The mean curvature vector H of M is defined by
H=
1 n
trace h.
For Lagrangian submanifolds, we have (cf. [10,12])
D X J Y = J ∇X Y ,
(2.3)
A J X Y = − J h( X , Y ) = A J Y X .
(2.4)
The above formulas immediately imply that h( X , Y ), J Z is totally symmetric. If we denote the curvature tensors of ∇ and D by R and R D , respectively, then the equations of Gauss and Codazzi are given by
R ( X , Y ) Z , W = A h(Y , Z ) X , W − A h( X , Z ) Y , W + c X , W Y , Z − X , Z Y , W ,
(∇ h)( X , Y , Z ) = (∇ h)(Y , X , Z ), where X , Y , Z , W (respectively,
(2.5) (2.6)
η and ξ ) are vector fields tangent (respectively, normal) to M; and ∇ h is defined by
(∇ h)( X , Y , Z ) = D X h(Y , Z ) − h(∇ X Y , Z ) − h(Y , ∇ X Z ).
(2.7)
For an orthonormal basis {e 1 , . . . , en } of T p M at a point p ∈ M, we put
A h BC = h(e B , e C ), J e A ,
A , B , C = 1, . . . , n .
It follows from (2.4) that A h BC = h BAC = h CA B .
(2.8)
2.2. Horizontal lift of Lagrangian submanifolds The following link between Legendre submanifolds and Lagrangian submanifolds is due to [14]. Case (i): CPn (4). Consider Hopf’s fibration π : S 2n+1 (1) → CPn (4). Then π is a Riemannian submersion. Consider S 2n+1 (1) as a Sasakian manifold with the canonical Sasakian structure with contact form η induced from the complex structure of Cn+1 . An n-dimensional submanifold N of S 2n+1 (1) is called a Legendre submanifold if η( X ) = 0 for every vector tangent to N. Given z ∈ S 2n+1 (1), the horizontal space at z is the orthogonal complement of iz with respect to the metric on S 2n+1 (1) induced from the metric on Cn+1 . Let ψ : N → CPn (4) be a Lagrangian immersion. Then there exists a covering map τ : Nˆ → N and a horizontal immersion ψˆ : Nˆ → S 2n+1 (1) such that ψ ◦ τ = π ◦ ψˆ . Thus each Lagrangian immersion can be lifted locally (or globally if N is simply-connected) to a Legendre immersion of the same Riemannian manifold. ˆ → S 2n+1 (1) is a Legendre immersion, then ψ = π ◦ φ : N → CPn (4) is again a Lagrangian immersion. Conversely, if φ : N Under this correspondence the second fundamental forms hψ and hφ of ψ and φ satisfy π∗ hφ = hψ . Moreover, hφ is horizontal with respect to π . +1 denote the complex pseudo-Euclidean (n + 1)-space endowed with the pseudo-Euclidean Case (ii): CHn (−4). Let C2n 1 metric
g 0 = −dz1 d z¯ 1 +
n +1
dz j d z¯ j .
j =2
Consider the anti-de Sitter spacetime
+1 H 12n+1 (−1) = z ∈ C2n : b1,n+1 ( z, z) = −1 1
with the canonical Sasakian structure (cf. [10, p. 245]). ¯ = 1}, where , is the Hermitian inner product on Cn+1 whose real part Put T z = {u ∈ Cn+1 : u , z = 0}, H 11 = {λ ∈ C: λλ 1
is g 0 . Then there is an H 11 -action on H 12n+1 (−1), z → λ z and at each point z ∈ H 12n+1 (−1), the vector ξ = −iz is tangent
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to the flow of the action. Since the metric g 0 is Hermitian, we have ξ, ξ = −1. The quotient space H 12n+1 (−1)/ ∼, under the identification induced from the action, is the complex hyperbolic space CHn (−4) with constant holomorphic sectional curvature −4, with the complex structure J induced from the complex structure J on Cn1+1 via π : H 12n+1 (−1) → CHn (4c ).
ˆ → N and Just like case (i), if ψ : N → CHn (−4) is a Lagrangian immersion, then there is an isometric covering map τ : N ˆ → H 2n+1 (−1) such that ψ ◦ τ = π ◦ φ . Thus every Lagrangian immersion can be lifted locally a Legendre immersion φ : N 1 (or globally if N is simply-connected) to a Legendre immersion. 2n + 1 ˆ →H (−1) is a Legendre immersion, then ψ = π ◦ φ : N → CHn (−4) is a Lagrangian immersion. Conversely, if φ : N 1 Similarly, under this correspondence the second fundamental forms hφ and hψ are related by π∗ hφ = hψ . Also, hφ is horizontal with respect to π . ˆ the Levi-Civita connections Assume M is a submanifold of S 2n+1 (1) ⊂ Cn+1 (or in H 12n+1 (−1) ⊂ Cn1+1 ). Denote by ∇ of Cn+1 (or of Cn1+1 ) and denote by L the corresponding immersion of M in Cn+1 (or of Cn1+1 ).
Let h be the second fundamental form of M in S 2n+1 (1) (or in H 12n+1 (−1)). Since S 2n+1 (1) and H 12n+1 (−1) are totally
umbilical with one as its mean curvature in Cn+1 and in Cn1+1 , respectively, we have
∇ˆ X Y = ∇ X Y + h( X , Y ) − ε L , where
(2.9) n+1
ε = 1 whenever the ambient space is C
; and
ε = −1 whenever the ambient space is
Cn1+1 .
3. Some lemmas We need the following lemmas for the proof of the main results.
˜ n (4c ) satisfying the equality case of (1.5) identically and let Lemma 3.1. Let M be a non-minimal Lagrangian submanifold of M {e 1 , . . . , en } be an adapted frame satisfying (1.6). Then we have (a) ∇e1 e 1 = 0; (b) e α λ = 0; (c) ∇eα e 1 = μe α with e 1 λ = (n − 1)λμ; for α = 2, . . . , n. Proof. Under the hypothesis, we have λ = 0. Moreover, we may choose e 2 such that
∇e1 e 1 = b1 e 2 .
(3.1)
By putting V = Span{e 3 , . . . , en }, it follows from (1.6), (2.3), (2.7), (3.1) and
(∇ h)(e 1 , e α , e 1 ) = (∇ h)(e α , e 1 , e 1 ) that
e1 λ n+1 for
J eα +
λ J ∇e1 e α − b1 h(e 2 , e α ) − h(e 1 , ∇e1 e α ) = e α J e 1 + λ J ∇eα e 1 − 2h(∇eα e 1 , e 1 ) n+1
α = 2, . . . , n. Thus, by comparing the J e 1 -component of (3.2), we find
n−1 e2 λ = b1 λ, n+1 V λ = 0, V ∈ V .
(3.2)
(3.3) (3.4)
Similarly, by comparing the J V -component of (3.2), we have
(n + 1)b1 J h(e 2 , e 2 )⊥ , n−1 λ λ (n + 1)b1 1 e1 λ ∇V e1 = V + J h(e 2 , V ) , n−1 λ λ ∇e2 e 1 =
1
e1 λ
e2 +
where h( X , Y )⊥ denotes the component of h( X , Y ) orthogonal to J e 1 , i.e.
h( X , Y )⊥ = h( X , Y ) − h( X , Y ), J e 1 J e 1 . After applying (3.1), (3.3)–(3.6), we find from
(∇ h)(e 2 , e j , e 1 ), J e j = (∇ h)(e j , e 2 , e 1 ), J e j
(3.5) (3.6)
B.-Y. Chen et al. / J. Math. Anal. Appl. 387 (2012) 139–152
143
that
0 = λ2 b 1 +
(n + 1)3 b1 h(e j , e j ), h(e 2 , e 2 )⊥ − h(e 2 , e j ), h(e 2 , e j ) (n − 1)2
(3.7)
for j = 3, . . . , n. Also, we obtain from
(∇ h)(e 1 , e j , e j ), J e 2 = (∇ h)(e j , e 1 , e j ), J e 2 ,
j = 3, . . . , n ,
that
0 = D e1 h(e j , e j ), J e 2 − 2
+
e1 λ
(n − 1)λ
n
∇e1 e j , e α h(e j , e α ), J e 2
α =2
h(e j , e j ), J e 2 −
(n + 1)b1 h(e j , e 2 ), h(e j , e 2 ) . (n − 2)λ
(3.8)
Furthermore, we derive from (∇ h)(e 1 , e 2 , e 2 ), J e 2 = (∇ h)(e 2 , e 1 , e 2 ), J e 2 that
0 = D e1 h(e 2 , e 2 ), J e 2 − 2
+
e1 λ
(n − 1)λ
n
α =2
(n + 3)b1 λ ∇e1 e 2 , e α h(e 2 , e α ), J e 2 + (n + 1)2
h(e 2 , e 2 ), J e 2 −
(n + 1)b1 h(e 2 , e 2 )⊥ , h(e 2 , e 2 )⊥ . (n − 1)λ
(3.9)
After taking the sum of (3.8) over j = 3, . . . , n and adding it with (3.9), we find
0=
n (n2 + n + 2)b1 λ (n + 1)b1 − h(e 2 , e α )⊥ , h(e 2 , e α )⊥ . (n − 1)λ (n + 1)2
(3.10)
α =2
On the other hand, by taking the sum of (3.7) over j = 3, . . . , n, we derive that
0=
n (n2 − 3n + 2)b1 λ (n + 1)b1 − h(e 2 , e α )⊥ , h(e 2 , e α )⊥ . 2 (n − 1)λ (n + 1)
(3.11)
α =2
Therefore, after combining (3.10) and (3.11), we obtain b1 = 0. Consequently, we have Lemma 3.1(a) by (3.1). Lemma 3.1(b) follows from (3.3), (3.4) and Lemma 3.1(a). Finally, Lemma 3.1(c) follows from (3.5), (3.6) and b1 = 0. 2 A foliation L on a Riemannian manifold M is called totally umbilical if every leaf of L is a totally umbilical submanifold of M. If, in addition, the mean curvature vector of every leaf is parallel in the normal bundle, then L is called a spheric foliation. If every leaf of L is a totally geodesic submanifold of M, then L is called a totally geodesic foliation. Lemma 3.2. Under the hypothesis and notations of Lemma 3.1, we have (i) the distribution D1 = Span{e 1 } is a totally geodesic foliation; (ii) the distribution D2 = Span{e 2 , . . . , en } is a spherical foliation; (iii) M is locally the warped product I × f N of an integral curve I of e 1 and a leaf of D2 , where f is the warping function. Proof. Since D1 is a one-dimensional distribution, it is a foliation. It follows from Lemma 3.1(a) that D1 is a totally geodesic foliation. The integrability of the distribution D2 follows from Lemma 3.1(c). Moreover, we conclude from Lemma 3.1(b) and Lemma 3.1(c) that D2 is a spherical foliation. Statement (iii) follows from statements (i) and (ii) and a result of Hiepko (cf. e.g. [10, p. 90]). 2 Let us choose a local coordinate system {t , u 2 , . . . , un } on the Lagrangian submanifold M such that e 1 = ∂∂t and {u 2 , . . . , un } are local coordinates of N. Lemma 3.3. Under the hypothesis and notations of Lemma 3.1, we have
dλ dt dμ dt
= (n − 1)λμ, = −c − μ2 −
(3.12) n λ2
(n + 1)2
.
(3.13)
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Proof. From the equation of Gauss we have
for
R (e α , e 1 )e 1 , e α = c + h(e 1 , e 1 ), h(e α , e α ) − h(e 1 , e α ), h(e 1 , e α )
α = 2, . . . , n. On the other hand, it follows from (1.6) and Lemma 3.1 that R (e α , e 1 )e 1 , e α = −e 1 μ − μ2 , n λ2 h(e 1 , e 1 ), h(e α , e α ) − h(e 1 , e α ), h(e 1 , e α ) = . (n + 1)2
(3.14)
(3.15) (3.16)
Let us put e 1 = ∂∂t . Then, we obtain (3.12) from Lemma 3.1(c). Moreover, by combining (3.14), (3.15) and (3.16), we obtain (3.13). 2 In view of Lemma 3.1(c), we may put
∇eα e β = −μδα β e 1 +
n
γ
α , β = 2, . . . , n ,
Γα β e γ ,
(3.17)
γ =2 γ
for some functions Γα β . 4. Lagrangian submanifolds of Cn satisfying the equality The following theorem completely classifies non-minimal Lagrangian submanifold of the complex Euclidean n-space Cn satisfying the equality case of (1.5). Theorem 4.1. Let M be a non-minimal Lagrangian submanifold of the complex Euclidean n-space Cn . Then
1
δ(n − 1) n(n − 1) H 2 .
(4.1)
4
The equality sign holds identically if and only if up to dilations and rigid motions, M is defined by
L (λ, u 2 , . . . , un ) =
(n + 1)e −iϕ φ(u 2 , . . . , un ), (n + 1)μ + iλ
where
ϕ (λ) = −
n+1 n
n
csc−1 (n + 1)c λ 1−n ,
(4.2)
2
μ(λ) = c 2 λ 1−n − (n + 1)−2 λ2
(4.3)
for some positive real number c, and φ(u 2 , . . . , un ) is a minimal Legendre submanifold of S 2n−1 (1). Proof. Let M be a non-minimal Lagrangian submanifold of the complex Euclidean n-space Cn . Assume that M satisfies the equality case of (4.1) identically. Then there exists an adapted frame {e 1 , . . . , en } which satisfies (1.6). From Lemma 3.1 and Lemma 3.3 we have
∇eα e 1 = μe α , α = 2, . . . , n, dλ = (n − 1)λμ, dt dμ dt Let
= −μ2 −
n λ2
(n + 1)2
.
(4.4) (4.5) (4.6)
ϕ (t ) be a function satisfying dϕ dt
= −λ.
(4.7)
Consider the map
φ = e iϕ e 1 .
(4.8)
Then φ, φ = 1. It follows from (1.6) and (4.7) that
∇˜ e1 φ = 0,
(4.9)
B.-Y. Chen et al. / J. Math. Anal. Appl. 387 (2012) 139–152
145
˜ is the Levi-Civita connection of Cn . Thus, φ is independent of t. Let L be the Lagrangian immersion of M in Cn . where ∇ Then it follows from (4.8) that e1 =
∂L = e −iϕ φ(u 2 , . . . , un ), ∂t
(4.10)
where u 2 , . . . , un are local coordinates of N. For each α ∈ {2, . . . , n}, we get from (1.6) and Lemma 3.1(c) that
iϕ ˜ iϕ ˜ φ∗ (e α ) = ∇eα φ = e ∇eα e 1 = e μ +
iλ n+1
eα .
(4.11)
Thus, from (1.6), (3.17), and (4.11), we find
∇˜ eβ φ∗ (e α ) = e iϕ μ +
iλ n+1
∇˜ eβ e α
n γ γ Γα β + ihα β φ∗ (e γ ) − μ2 + = γ =2
=
λ2 δα β φ (n + 1)2
n γ γ Γα β + ihα β φ∗ (e γ ) − φ∗ (e α ), φ∗ (e β ) φ
(4.12)
γ =2
α , β = 2, . . . , n. Since M is a Lagrangian submanifold in Cn , (4.8) and (4.11) imply that iφ is perpendicular to each tangent space of M. Hence, φ is a horizontal immersion in the unit hypersphere S 2n−1 (1) ⊂ Cn . Moreover, it follows from (4.12) that the second fundamental form of φ is the original second fundamental form of M respect to the second factor N of the warped product I × f N. Hence, φ is a minimal horizontal immersion in S 2n−1 (1). Therefore, φ is a horizontal lift of a minimal Lagrangian immersion in CPn−1 (4). By direct computation, we find for
∇˜ e1 L − for
n+1
(n + 1)μ + iλ
e1
= ∇˜ eα L −
n+1
(n + 1)μ + iλ
e1
=0
α = 2, . . . , n. Thus, up to translations the Lagrangian immersion L is given by
(n + 1)e −iϕ φ(u 2 , . . . , un ), L= (n + 1)μ + iλ
where φ is a horizontal minimal immersion in S 2n−1 (1) and λ,
dλ dt
dϕ
= (n − 1)λμ,
dt
= −λ,
dμ dt
= −μ2 −
(4.13)
(4.14)
ϕ , μ satisfy
n λ2
(n + 1)2
.
(4.15)
From (4.15) we find
d
dt
λ
2 n −1
λ2 μ + (n + 1)2
2
= 0,
2
(4.16)
2
λ which implies that λ n−1 (μ2 + (n+ ) = c 2 for some positive number c. Thus 1)2
2
μ = ± c 2 λ 1−n −
λ2 , (n + 1)2
(4.17)
for some positive real number c. Replacing e 1 by −e 1 if necessary, we have
2
μ = c 2 λ 1−n −
λ2 . (n + 1)2
(4.18)
Now, it follows from (4.15) and (4.18) we get
dϕ dλ
=
1
2 (1 − n) c 2 λ 1−n −
. λ2 (n+1)2
(4.19)
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Thus we obtain
ϕ =b−
n+1 n
n
csc−1 (n + 1)c λ 1−n .
(4.20)
Therefore, after applying a suitable rotation of Cn , we get (4.2) and (4.3). The converse can be verified by straightforward computation. 2 A Lagrangian submanifold of Cn without totally geodesic points is called a Lagrangian H -umbilical submanifold if its second fundamental form takes the following simple form (cf. [5,6]):
h(¯e 1 , e¯ 1 ) = ϕ J e¯ 1 ,
h(¯e j , e¯ j ) = μ J e¯ 1 ,
h(¯e 1 , e¯ j ) = μ J e¯ j ,
h(¯e j , e¯ k ) = 0,
j > 1,
2 j = k n
(4.21)
for some functions ϕ , μ with respect to a suitable orthonormal local frame field {¯e 1 , . . . , e¯ n }. Such submanifolds are the simplest Lagrangian submanifolds next to the totally geodesic ones. Let G : N n−1 → En be an isometric immersion of a Riemannian (n − 1)-manifold into the Euclidean n-space En and let F : I → C∗ be a unit speed curve in C∗ = C − {0}. We may extend G : N n−1 → En to an immersion of I × N n−1 into Cn as
F ⊗ G : I × N n−1 → C ⊗ En = Cn ,
(4.22)
where ( F ⊗ G )(s, p ) = F (s) ⊗ G ( p ) for s ∈ I , p ∈ N extensor of G via F . The following result was proved in [6].
n−1
. This extension F ⊗ G of G via tensor product is called the complex
Proposition 4.1. Let ι : S n−1 → En be the inclusion of a hypersphere of Em centered at the origin. Then every complex extensor φ = F ⊗ ι of ι via a unit speed curve F : I → C∗ is a Lagrangian H -umbilical submanifold of Cn unless F is contained in a line through the origin. For F ⊗ ι, let us choose a unit vector field e¯ 1 tangent to the first factor and e¯ 2 , . . . , e¯ n tangent to the second factor of I × S n−1 . If we put F (s) = e iζ (s) and F (s) = r (s)e iθ(s) , then it follows from [6] that the second fundamental form of F ⊗ ι satisfies (6.1) with
ϕ = ζ (s) = κ ,
μ=
F , iF = θ (s), F , F
(4.23)
where κ is the curvature function of F . By applying (4.21), (4.23) and Theorem C, we conclude that if the unit speed curve F : I → C satisfies
κ (s) = (n + 1)θ (s) = 0, s ∈ I , then the complex extensor F ⊗ ι : I × S case of (1.5). Remark 4.1. The curve
γ (λ) =
(4.24) n−1
→ C is a non-minimal Lagrangian submanifold of C which verifies the equality n
n
(n+1)e iϕ (n+1)μ+iλ in C defined by (4.2) and (4.3) satisfies condition (4.24).
Remark 4.2. When φ(u 2 , . . . , un ) in (4.2) is a totally geodesic Legendre submanifold of S n−1 (1), then (4.2) reduces to the complex extensor given above. 5. Lagrangian submanifolds of CPn (4) satisfying the equality Theorem 5.1. Let M be a non-minimal Lagrangian submanifold of the complex projective n-space CPn (4). Then
δ(n − 1)
n − 1 2 nH + 4 . 4
(5.1)
The equality sign holds identically if and only if M is congruent to the Lagrangian submanifold defined by the composition π ◦ L, where π : S 2n+1 (1) → CPn (4) is the Hopf fibration and
L (t , u 2 , . . . , un ) =
(i(n + 1)−1 λ − μ)e −niθ , , 1 + μ2 + (n + 1)−2 λ2 1 + μ2 + (n + 1)−2 λ2
where λ(t ), μ(t ) and θ(t ) satisfy
e −iθ φ
(5.2)
B.-Y. Chen et al. / J. Math. Anal. Appl. 387 (2012) 139–152
dθ dt dλ
=−
λ n+1
,
147
λ = 0,
= (n − 1)λμ,
dt dμ
= −1 − μ2 −
dt
n λ2
(5.3)
(n + 1)2
and φ is a minimal Legendre immersion in S 2n−1 (1). Proof. Let M be a non-minimal Lagrangian submanifold of the complex projective n-space CPn (4). Assume that M satisfies the equality case of (5.1) identically. Then there exists an adapted frame {e 1 , . . . , en } which satisfies (1.6). From Lemma 3.1 and Lemma 3.3 we have
∇eα e 1 = μe α , α = 2, . . . , n, dλ = (n − 1)λμ,
(5.4) (5.5)
dt
dμ
= −1 − μ2 −
dt
n λ2
(n + 1)2
(5.6)
.
Let L denote the immersion in Cn+1 associated with the horizontal lift of the Lagrangian immersion of M in CPn (4) via Hopf’s fibration π : S 2n+1 (1) → CPn (4) and let θ(t ) be a function on M satisfying
dθ dt
=−
λ . n+1
(5.7)
Consider the map:
ξ=
eniθ (e 1 − (μ + ni+λ1 ) L )
2
λ 1 + μ2 + (n+ 1 )2
(5.8)
.
˜ eα ξ = 0, α = 2, . . . , n, where ∇˜ is the Levi-Civita connection Then ξ, ξ = 1. Also, it follows from (1.6) and (5.4) that ∇ ˜ e1 ξ = 0. Hence, ξ is a constant unit vector of Cn+1 . Moreover, it follows from Lemma 3.1(a), (1.6), and (5.5)–(5.7) that ∇ in Cn+1 . Let us put e iθ ( L + (μ − ni+λ1 )e 1 ) φ= . λ2 1 + μ2 + (n+ 2 1)
(5.9)
˜ e1 φ = 0. Thus, φ is independent of t. Now, by combining Then, by applying Lemma 3.1(a), (1.6), and (5.5)–(5.7), we find ∇ (5.8) and (5.9), we get L=
e −iθ φ − e −niθ (μ − ni+λ1 )ξ
2
λ 1 + μ2 + (n+ 1 )2
(5.10)
.
Since φ is orthogonal to ξ , iξ , we obtain (5.2) after choosing ξ = (0, . . . , 0, 1) ∈ Cn+1 . It follows from (1.6), (5.4) and (5.9) that
∇ˆ eα φ = e iθ 1 + μ2 + for
λ2 eα (n + 1)2
(5.11)
α = 2, . . . , n. Thus, by applying (3.17) and (5.11) we find
λ2 ∇ˆ eβ ∇ˆ eα φ = e iθ 1 + μ2 + (∇ˆ eβ e α ) (n + 1)2
n γ λ2 γ iθ 2 Γα β + ihα β φ∗ (e γ ) − φ∗ (e α ), φ∗ (e β ) φ = e 1+μ + (n + 1)2 γ =2
(5.12)
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B.-Y. Chen et al. / J. Math. Anal. Appl. 387 (2012) 139–152
for α , β = 2, . . . , n. Hence, φ is a horizontal immersion in the unit hypersphere S 2n−1 (1) ⊂ Cn . Moreover, it follows from (5.12) that the second fundamental form of φ is a scalar multiple of the original second fundamental form of M restricted to the second factor of the warped product I × f N. Hence, φ is a minimal horizontal immersion in S 2n−1 (1). The converse can be verified by direct computation. 2 Remark 5.1. Theorem 5.1 extends the main result of [3] (see also [1,2]). 6. Lagrangian submanifolds of CH n (−4) satisfying the equality Finally, we completely classify non-minimal Lagrangian submanifolds of the complex hyperbolic n-space CHn (−4) satisfying the equality case of inequality (1.5). Theorem 6.1. Let M be a non-minimal Lagrangian submanifold of the complex hyperbolic n-space CHn (−4). Then
δ(n − 1)
n − 1 2 nH − 4 . 4
(6.1)
The equality sign holds identically if and only if M is congruent to the Lagrangian submanifold defined by the composition π ◦ L, where π : H 12n+1 (−1) → CHn (−4) is the Hopf fibration and L is one of the following immersions: (a) the immersion given by
L (t , u 2 , . . . , un ) =
(i(n + 1)−1 λ − μ)e −niθ , , 1 − μ2 − (n + 1)−2 λ2 1 − μ2 − (n + 1)−2 λ2 e −iθ φ
(6.2)
where λ(t ), μ(t ) and θ(t ) satisfy
dθ dt dλ dt dμ dt
=−
λ n+1
λ = 0,
,
1 > μ2 + (n + 1)−2 λ2 ,
= (n − 1)λμ, = 1 − μ2 −
n λ2
(6.3)
(n + 1)2
and φ is a minimal Legendre immersion in H 12n−1 (−1); (b) the immersion given by
L (t , u 2 , . . . , un ) =
e −iθ φ (i(n + 1)−1 λ − μ)e −niθ , , μ2 + (n + 1)−2 λ2 − 1 μ2 + (n + 1)−2 λ2 − 1
(6.4)
where λ(t ), μ(t ) and θ(t ) satisfy
dθ dt dλ dt dμ dt
=−
λ , n+1
λ = 0,
= (n − 1)λμ, = 1 − μ2 −
1 < μ2 + (n + 1)−2 λ2 , n λ2
(6.5)
(n + 1)2
and φ is a minimal Legendre immersion in S 12n−1 (1); (c) the immersion given by
L (t , u 2 , . . . , un ) =
2i 1 −1 e n−1 tan (tanh( 2 (n−1)t )) 1
cosh n−1 ((n − 1)t )
t
+
2
cosh 1−n (n − 1)t e
w+
i 2
i φ, φ + i, φ, w + φ, φ 2
2i tan−1 (tanh( 12 (n−1)t ))
dt (1, 0, . . . , 0, 1) ,
(6.6)
0
where φ(u 2 , . . . , un ) is a minimal Lagrangian immersion in Cn−1 and up to constants w (u 2 , . . . , un ) is the unique solution of the PDE system:
∂w ∂φ = i ,φ , ∂ uα ∂ uα
α = 2, . . . , n .
(6.7)
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149
Proof. Let M be a non-minimal Lagrangian submanifold of the complex hyperbolic n-space CHn (−4). Assume that M satisfies the equality case of (6.1) identically. Then there exists an adapted frame {e 1 , . . . , en } which satisfies (1.6). From Lemma 3.1 and Lemma 3.3 we have
∇eα e 1 = μe α , α = 2, . . . , n, dλ = (n − 1)λμ,
(6.8) (6.9)
dt
dμ
= 1 − μ2 −
dt
n λ2
(n + 1)2
(6.10)
.
Let L be the immersion in Cn1+1 associated with the horizontal lift of the Lagrangian immersion in CHn (−4) via Hopf’s fibration
dθ dt
π : H 12n+1 (−1) → CHn (−4) and let θ(t ) be a function satisfying =−
λ n+1
(6.11)
.
It follows from (6.9) and (6.10) that
d
dt
1 − μ2 −
λ2 (n + 1)2
= −2μ 1 − μ2 −
λ2 . (n + 1)2
(6.12) 2
λ Since this first order differential equation for y (t ) = 1 − μ2 − (n+ has a unique solution for each given initial condition, 1)2 every solution is either vanishing identically or nowhere vanishing. Now, we divide the proof into three cases. 2
λ > 0. Consider the map: Case (i): 1 − μ2 − (n+ 1)2
η=
eniθ (e 1 − (μ + ni+λ1 ) L )
2
λ 1 − μ2 − (n+ 1 )2
(6.13)
.
˜ eα η = 0, α = 2, . . . , n, where ∇˜ is the Levi-Civita connection Then η, η = 1. Also, it follows from (1.6) and (6.8) that ∇ n+1 ˜ e1 ξ = 0 as well. Hence, η is a constant space-like of C1 . Moreover, it follows Lemma 3.1(a), (1.6), and (6.9)–(6.11) that ∇
unit vector in Cn1+1 . Let us put
e iθ ((μ − ni+λ1 )e 1 − L ) φ= . λ2 1 − μ2 − (n+ 1 )2
(6.14)
˜ e1 φ = 0. Thus, φ is independent of t. Now, by combining Then, by applying Lemma 3.1(a), (1.6), and (6.9)–(6.11), we find ∇ (6.13) and (6.14), we get L=
e −niθ (μ − ni+λ1 )η − e −iθ φ
2
λ 1 − μ2 − (n+ 1 )2
Since φ is orthogonal to
.
(6.15)
η, iη, we find (6.2) after choosing η = (0, . . . , 0, 1) ∈ Cn1+1 . Now, by applying similar arguments as
given in the proof of Theorem 5.1, we conclude that φ is a minimal Legendre immersion in H 12n−1 (−1). This gives case (i) of the theorem. 2
λ < 0. In this case, we consider the following maps: Case (ii): 1 − μ2 − (n+ 1)2
η=
eniθ (e 1 − (μ + ni+λ1 ) L )
2
μ2 + (n+λ 1)2 − 1
e iθ ((μ − ni+λ1 )e 1 − L ) φ= 2 μ2 + (n+λ 1)2 − 1
,
(6.16)
(6.17)
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B.-Y. Chen et al. / J. Math. Anal. Appl. 387 (2012) 139–152
instead. Then, after applying similar arguments as case (i), we may conclude that η is a constant unit time-like vector and φ is independent of t which is orthogonal to η , iη . Moreover, we may prove that φ is a minimal Legendre immersion in S 2n−1 (1). Thus, after choosing η = (1, 0, . . . , 0) we obtain case (ii) of the theorem. 2
λ = 0. By solving (6.9) and (6.10), we find Case (iii): 1 − μ2 − (n+ 1)2
λ = (n + 1) sech (1 − n)t ,
μ = tanh (1 − n)t .
(6.18)
It follows from ∇e1 e 1 = 0, (1.6) and (6.18) that the horizontal lift L of the Lagrangian immersion of M in CH (−4) satisfies n
L tt − i(n + 1) sech (1 − n)t L t − L = 0.
(6.19)
Solving this second order differential equation gives
2i 1 1 −1 L = cosh 1−n (n − 1)t e n−1 tan (tanh( 2 (n−1)t )) A + ρ (t )ζ ,
(6.20)
where A (u 2 , . . . , un ) and ζ (u 2 , . . . , un ) are Cn1+1 -valued functions and
t
ρ (t ) =
2 1 −1 cosh n−1 (n − 1)t e 2i tan (tanh( 2 (n−1)t )) dt .
(6.21)
0
By direct computation we have
t
ρ (t ) =
3−n
cosh n−1 (n − 1)t dt +
i 2
2
cosh n−1 (n − 1)t − 1 .
(6.22)
0
On the other hand, it follows from Lemma 3.1(c), (1.6) and (6.18) that
L tu α = tanh (1 − n)t + i sech (1 − n)t
L uα ,
By substituting (6.20) into (6.23), we find ζu α = 0, Since L , L = −1, it follows from (6.20) that
α = 2, . . . , n .
(6.23)
α = 2, . . . , n. Thus ζ is a constant vector in
Cn1+1 .
2 − cosh n−1 (n − 1)t = A , A + 2 A , ρ (t )ζ + ρ (t )ζ, ρ (t )ζ . Since A , A is independent of t and
(6.24)
ρ (0) = 0, (6.24) implies that
A , A = −1.
(6.25)
Hence, we get from (6.24) that
2 A , ρ (t )ζ +
2
ρ (t )ζ, ρ (t )ζ = 1 − cosh n−1 (n − 1)t .
(6.26)
Differentiating (6.26) with respect to t yields
A , ρ (t )ζ +
n −3
ρ (t )ζ, ρ (t )ζ = − sinh (n − 1)t sech n−1 (n − 1)t .
(6.27)
Thus, by taking t = 0, we find from (6.21) and (6.27) that
A , ζ = 0.
(6.28)
By applying (6.22), (6.26) and (6.28), we find
2
ρ (t )ζ, ρ (t )ζ = 1 − cosh n−1 (n − 1)t
1 + A , iζ .
(6.29)
Thus, after differentiating (6.29) with respect to t twice, we obtain
ρ (t )ζ, ρ (t )ζ + ρ (t )ζ, ρ (t )ζ = 2 − n − cosh (n − 1)t sech
2(n−2) n −1
(n − 1)t 1 + A , iζ .
(6.30)
Thus, at t = 0, (6.30) implies that
ζ, ζ = (1 − n) 1 + A , iζ .
(6.31)
B.-Y. Chen et al. / J. Math. Anal. Appl. 387 (2012) 139–152
151
On the other hand, it follows from (6.20) and (6.21) that
2i 1 1 −1 L t = cosh 1−n (n − 1)t e n−1 tan (tanh( 2 (n−1)t )) ρ (t )ζ + tanh (n − 1)t + i sech (n − 1)t A + ρ (t )ζ .
(6.32)
Since L is a Legendre immersion, we have L t , iL = 0. Thus we derive from (6.20) and (6.32) that
0=
ρ (t )ζ, i A + ρ (t )ζ + tanh (n − 1)t + i sech (n − 1)t
Since ρ (0) = 0, ρ (0) = 1 and
A + ρ (t )ζ , i A + ρ (t )ζ .
(6.33)
A , A = −1, (6.33) implies that
ζ, i A = 1.
(6.34)
By combining this with (6.31), we conclude that ζ, ζ = 0. Thus, ζ is a constant light-like vector. Without loss of generality, we may put
ζ = (1, 0, . . . , 0, 1).
(6.35)
Let us put
A = (a1 + ib1 , a2 + ib2 , . . . , an+1 + ibn+1 ).
(6.36)
Then it follows from (6.28) and (6.34)–(6.36) that an+1 = a1 and bn+1 = b1 − 1. Therefore
A = a1 + ib1 , a2 + ib2 , . . . , a1 + i(b1 − 1) .
(6.37)
Now, by using (6.25) and (6.37), we find
b1 =
1 2
φ, φ + 1.
(6.38)
By combining (6.37) and (6.38), we obtain
A=
w+
i 2
i φ, φ + i, φ, w + φ, φ ,
(6.39)
2
where w = a1 and φ = (a2 + ib2 , . . . , an + ibn ). Since L u α , iL = 0, it follows from (6.20) that
A u α , i A + A u α , iρ (t )ζ = 0,
α = 2, . . . , n .
(6.40)
Also, it follows from (6.35) and (6.39) that A u α , iρ (t )ζ = 0. On the other hand, (6.39) implies that
∂w ∂φ A uα , i A = + , iφ . ∂ uα ∂ uα
(6.41)
Thus the function w satisfies
∂w ∂φ = i ,φ . ∂ uα ∂ uα
(6.42)
Consequently, we obtain from (6.20), (6.35) and (6.37) that
i i L = H (t ) w + φ, φ + i, φ, w + φ, φ + ρ (t )(1, 0, . . . , 0, 1) , 2
2
(6.43)
where
H (t ) =
2i 1 −1 e n−1 tan (tanh( 2 (n−1)t )) 1
cosh n−1 ((n − 1)t )
(6.44)
,
and w (u 2 , . . . , un ) is a function satisfying (6.42). It follows from (6.43) that
L u α = H (t ) w u α + iφu α , φ , φu α , w u α + iφu α , φ . 2
(6.45)
Thus φu α , φu β = cosh n−1 ((n − 1)t ) L u α , L u β , which shows that φ is an immersion in Cn−1 . Also, we find from (6.44) and L u α , iL u β = 0 that φu α , iφu β = 0. Hence, φ is Lagrangian.
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B.-Y. Chen et al. / J. Math. Anal. Appl. 387 (2012) 139–152
Moreover, it follows from (6.43) that
i i e 1 = H (t ) w + φ, φ + i, φ, w + φ, φ + ρ (t )(1, 0, . . . , 0, 1) + H (t )ρ (t )(1, 0, . . . , 0, 1).
2
2
(6.46)
From (6.43) and (6.46) we find
(1, 0, . . . , 0, 1) = For
e1 H (t )ρ (t )
+
H (t ) H 2 (t )
L.
(6.47)
α , β = 2, . . . , n, we obtain from (1.6), (2.9), (3.17) and (6.18) that n γ γ Γα β + ihα β L uk + δα β L . ∇ˆ ea e β = δα β i sech (1 − n)t − tanh (1 − n)t e 1 +
(6.48)
γ =2
Therefore, (6.45), (6.47) and (6.48) give
φα β =
n γ γ Γα β + ihα β φuk .
(6.49)
γ =2
Hence, we may conclude from (1.6), (6.48) and (6.49) that φ is a minimal Lagrangian immersion in Cn−1 . Consequently, we obtain case (iii) of the theorem. The converse can be verified by direct computations. 2 References [1] J. Bolton, F. Dillen, J. Fastenakels, L. Vrancken, A best possible inequality for curvature-like tensor fields, Math. Inequal. Appl. 12 (2009) 663–681. [2] J. Bolton, C. Rodriguez Montealegre, L. Vrancken, Characterizing warped product Lagrangian immersions in complex projective space, Proc. Edinb. Math. Soc. 51 (2008) 1–14. [3] J. Bolton, L. Vrancken, Lagrangian submanifolds attaining equality in the improved Chen’s inequality, Bull. Belg. Math. Soc. Simon Stevin 14 (2007) 311–315. [4] B.-Y. Chen, Some pinching and classification theorems for minimal submanifolds, Arch. Math. 60 (1993) 568–578. [5] B.-Y. Chen, Interaction of Legendre curves and Lagrangian submanifolds, Israel J. Math. 99 (1997) 69–108. [6] B.-Y. Chen, Complex extensors and Lagrangian submanifolds in complex Euclidean spaces, Tohoku Math. J. 49 (1997) 277–297. [7] B.-Y. Chen, Some new obstruction to minimal and Lagrangian isometric immersions, Jpn. J. Math. 26 (2000) 105–127. [8] B.-Y. Chen, Ideal Lagrangian immersions in complex space forms, Math. Proc. Cambridge Philos. Soc. 128 (2000) 511–533. [9] B.-Y. Chen, Riemannian submanifolds, in: F. Dillen, L. Verstraelen (Eds.), Handbook of Differential Geometry, vol. I, North-Holland, Amsterdam, 2000, pp. 187–418. [10] B.-Y. Chen, Pseudo-Riemannian Geometry, δ -Invariants and Applications, World Scientific, Hackensack, NJ, 2011. [11] B.-Y. Chen, F. Dillen, Optimal general inequalities for Lagrangian submanifolds in complex space forms, J. Math. Anal. Appl. 379 (2011) 229–239. [12] B.-Y. Chen, K. Ogiue, On totally real submanifolds, Trans. Amer. Math. Soc. 193 (1974) 257–266. [13] T. Oprea, Chen’s inequality in the Lagrangian case, Colloq. Math. 108 (2007) 163–169. [14] H. Reckziegel, Horizontal lifts of isometric immersions into the bundle space of a pseudo-Riemannian submersion, Lecture Notes in Math. 1156 (1985) 264–279.