A construction of special Lagrangian submanifolds by generalized perpendicular symmetries
AA CONSTRUCTION OF SPECIAL LAGRANGIANSUBMANIFOLDS BY GENERALIZED PERPENDICULARSYMMETRIES
AKIFUMI OCHIAI
Abstract.
We show a method to construct a special Lagrangian submanifold L (cid:48) from a given special Lagrangian submanifold L in a Calabi-Yau manifoldwith the use of generalized perpendicular symmetries. We use moment mapsof the actions of Lie groups, which are not necessarily abelian. By our method,we construct some non-trivial examples in non-flat Calabi-Yau manifolds T ∗ S n which equipped with the Stenzel metrics. Introduction
In 1982, Harvey and Lawson introduced a special class of submanifolds, namelycalibrated submanifolds in their paper [3]. Calibrated submanifolds has a strongproperty that they realize volume minimizing submanifolds in the homologicalclass. Particularly, in Calabi-Yau manifolds calibrated submanifolds which havehalf-dimensions are defined and they are called special Lagrangian submanifolds.Because special Lagrangian submanifolds play an important role for understandingmirror symmetries and the SYZ-conjecture, many mathematicians pay attention totheir constructions and singularities.Let us review the history of constructions of special Lagrangian submanifolds,regarding their ambient spaces and methods of constructions. At first C n waschosen for an ambient space and in there various examples and methods of con-structing special Lagrangian submanifolds were given by Joyce in a series of hispapers [8]–[12]. On the other hand, Stenzel gave examples of non-flat Calabi-Yaustructures on the conormal bundles over compact rank one symmetric spaces. Nextspecial Lagrangian submanifolds are constructed in those spaces (first in T ∗ S n , andrecently in T ∗ C P n ).One of the useful method of constructing special Lagrangian submanifolds iscalled moment map techniques which were introduced by Joyce in [11]. This methodneeds large symmetries, and by using these symmetries we can reduce PDEs forbeing special Lagrangian submanifolds to ODEs on the orbit spaces. Using thismethod, Joyce constructed special Lagrangian submanifolds in C n ( ∼ = T ∗ R n ) invari-ant under a subgroup of SU ( n ). With this method special Lagrangian submanifoldswere also studied in T ∗ S n by Anciaux [1], Ionel and Min-Oo [7], Hashimoto andSakai [5], Hashimoto and Mashimo [4], and in T ∗ C P n by Arai and Baba [2]. All ofthese examples were cohomogeneity one. Mathematics Subject Classification.
Key words and phrases. special Lagrangian submanifold, Calabi-Yau manifold, minimal sub-manifold, moment map. a r X i v : . [ m a t h . DG ] J u l AKIFUMI OCHIAI
Another method was introduced by Harvey and Lawson [3] which is called bundletechniques. With the use of this method, Karigiannis and Min-Oo [13] constructedspecial Lagrangian submanifolds in T ∗ S n , and Ionel and Ivey [6] in T ∗ C P n .Aside from these two typical methods, Joyce [11] showed a way to construct aspecial Lagrangian submanifold L (cid:48) in C n from another given special Lagrangiansubmanifold L by using actions of an abelian group which acts perpendicularly to L . This method has advantage that we need not deal with the PDE for L (cid:48) tobe a special Lagrangian submanifold (it is “already achieved” by the given specialLagrangian submanifold L ), and that large symmetries are not necessarily needed.In this paper we generalize this Joyce’s result above using “perpendicular sym-metries” in three points. Firstly we generalize ambient spaces to general Calabi-Yaumanifolds from C n . Secondly we do not assume the commutativity of Lie groups.Thirdly we generalize the condition that the group acts perpendicularly to a givenspecial Lagrangian submanifold. By this method we also construct non-trivial ex-amples of special Lagrangian submanifolds in Calabi-Yau manifolds T ∗ S n equippedwith the Stenzel metrics.The method to construct special Lagrangian submanifolds in this paper is sum-marized as follows: Let ( M, I, ω,
Ω) be a connected Calabi-Yau manifold and H aconnected Lie group which acts on M preserving I . Let h and h ∗ be the Lie algebraof H and its dual respectively. Assume the H -action is Hamiltonian, i.e. ( M, ω, H )has a moment map µ : M → h ∗ . Let L be a special Lagrangian submanifold of( M, I, ω,
Ω) and Z ( h ∗ ) the center of h ∗ . Suppose that for c ∈ Z ( h ∗ ), V c is a sub-manifold of M which satisfies V c ⊂ µ − ( c ) ∩ L and dim H + dim V c = dim M .Assume that the actions of H are “(generalized) perpendicular actions” for V c (notnecessarily for whole of L ). Then H · V c is a special Lagrangian submanifold.Konno [14] showed in general Calabi-Yau manifolds a method of constructingLagrangian mean curvature flows by using perpendicular actions of abelian groupsfor given special Lagrangian submanifolds, and constructed some examples. Thispaper is inspired from the study by Konno.2. Preliminaries
In this section, we review some fundamental facts about Calabi-Yau manifolds,their special Lagrangian submanifolds, group actions, and moment maps.2.1.
Special Lagrangian submanifolds.
We begin with the definition of La-grangian submanifolds in symplectic manifolds.Let (
M, ω ) be a symplectic manifold. A submanifold L of ( M, ω ) is isotropic if ω | L ≡
0. If an isotropic submanifold L is of half-dimension of dim M , it is called a Lagrangian submanifold .Next we see the definition of special Lagrangian submanifolds. It is a particularsubmanifold of a Calabi-Yau manifold which is defined as follows:
Definition 2.1.
A Calabi-Yau manifold is a quadruple ( M, I, ω, Ω) such that ( M, I ) is a complex manifold equipped with a K¨ahler form ω and a holomorphic volumeform Ω which satisfy the following relation: ω n n ! = ( − n ( n − (cid:18) √− (cid:19) n Ω ∧ Ω . CONSTRUCTION OF SPECIAL LAGRANGIAN SUBMANIFOLDS 3 If L is an oriented Lagrangian submanifold of a Calabi-Yau manifold ( M, I, ω,
Ω),there exists a function θ : L → R / π Z , which is called the Lagrangian angle satis-fying ι ∗ Ω = e √− θ vol ι ∗ g . Here g is the K¨ahler metric, ι : L → M is an embedding, and vol ι ∗ g is the volumeform on L with respect to the induced metric ι ∗ g . Even if L is not orientable, wecan locally define the Lagrangian angle with the formula above. With the use ofthe Lagrangian angle θ of a Lagrangian submanifold L , the mean curvature vector H p at p ∈ L is expressed as follows: H p = I ι ( p ) ( ι ∗ p ( ∇ ι ∗ g θ ) p ) ∈ T ⊥ ι ( p ) ι ( L ) , where ∇ ι ∗ g θ is the gradient of the function θ with respect to the induced metric ι ∗ g .The definition of a special Lagrangian submanifold is given by the following: Definition 2.2.
Let ( M, I, ω, Ω) be a Calabi-Yau manifold. A special Lagrangiansubmanifold of ( M, I, ω, Ω) is a Lagrangian submanifold such that its Lagrangianangle is constant θ ≡ θ . θ is called the phase of the special Lagrangian submani-fold. From the formula of the mean curvature vector above, we can see that a specialLagrangian submanifold is a minimal submanifold. More strongly it is known thata special Lagrangian submanifold is homologically volume minimizing.2.2.
Group actions and moment maps.
In this subsection we review the fun-damental notions of group actions and moment maps.Let H be a Lie group which acts on M . We denote the translation of h ∈ H by L h : M → M . For each p ∈ M , the orbit and the isotropy subgroup at p aredenoted by H · p and H p respectively.Letting h denote the Lie algebra of H , any ξ ∈ h induces a fundamental vectorfield ξ on M , defined as follows: ξ p = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 exp( tξ ) p ( p ∈ M )where exp( tξ ) denotes the 1-parameter subgroup of H associated to ξ . H acts on h ∗ by the coadjoint action :Ad ∗ h : h ∗ → h ∗ , where h ∈ H , and for c ∈ h ∗ , Ad ∗ h c is defined as follows: (cid:104) Ad ∗ h c, ξ (cid:105) = (cid:104) c, Ad h ξ (cid:105) ( ξ ∈ h ) . Here (cid:104)· , ·(cid:105) is the pairing of h and h ∗ . We call Z ( h ∗ ) = { c ∈ h ∗ | Ad ∗ h c = c, h ∈ H } the center of h ∗ . If H is abelian, then Z ( h ∗ ) = h ∗ holds. Definition 2.3.
Let H be a Lie group acting on a symplectic manifold ( M, ω ) . Amoment map µ : M → h ∗ is an H -equivariant map that satisfies for any ξ ∈ h thefollowing: − i ( ξ ) ω = d (cid:104) µ ( · ) , ξ (cid:105) , where i is the interior product. AKIFUMI OCHIAI
If (
M, ω, H ) has a moment map, the H -action is called Hamiltonian . A Hamil-tonian action preserves ω . For each p ∈ µ − ( c ), c ∈ h ∗ , the orbit H · p is isotropicif and only if c ∈ Z ( h ∗ ).3. Transformations of holomorphic volume forms
In this section, we retain the notation as in Section 2. We show a formula (Propo-sition 3.2) corresponding to transformations of holomorphic volume forms L ∗ h Ω. Weuse this formula to calculate the Lagrangian angle of a Lagrangian immersion whichwe finally construct in Theorem 4.1.Let (
M, I ) be a complex manifold and Ω a holomorphic volume form on M . Let H be a Lie group which acts on M preserving I . Then the map( L h ) ∗ : A k ( M ) C → A k ( M ) C ; ω (cid:55)→ L ∗ h ω preserves types of complex differential k -forms ( k ∈ N ), where A k ( M ) C is thecomplex vector space which consists of all complex k -forms on M . Hence L ∗ h Ωis an ( n, f h that satisfies L ∗ h Ω = f h Ω.Next we introduce a Calabi-Yau structure into a complex manifold, and assumethat an H -action preserves the K¨ahler structure. Then we can see that the holo-morphic function f h satisfies | f h | ≡ Proposition 3.1.
Let ( M, I, ω, Ω) be a n -dimensional Calabi-Yau manifold and H a Lie group which acts on M preserving I and ω . Then f h satisfies | f h | ≡ on M .Proof. The quadruple ( M, ( L h ∗ ) − ◦ I ◦ ( L h ∗ ) , L ∗ h ω, L ∗ h Ω) is also a Calabi-Yau man-ifold for any h ∈ H , since H preserves I and ω . Therefore, we have( − n ( n − (cid:18) √− (cid:19) n Ω ∧ Ω = ω n n ! = ( L ∗ h ω ) n n !=( − n ( n − (cid:18) √− (cid:19) n L ∗ h Ω ∧ L ∗ h Ω = ( − n ( n − (cid:18) √− (cid:19) n | f h | Ω ∧ Ω . Comparing the both sides, we obtain | f h | ≡ (cid:3) By Proposition 3.1 we know the following: Because a holomorphic function whichhas a constant norm on a connected space has to be constant, f h ≡ const . ∈ U (1)on a connected Calabi-Yau manifold. Therefore we can define a map c : H → U (1); h (cid:55)→ c ( h ) = c h := f h .The map c is a homomorphism between Lie groups. In fact for h , h ∈ H , wehave c h c h Ω = L ∗ h ( L ∗ h Ω) = L ∗ h h Ω = c h h Ω . Therefore c h h = c h c h = c h c h , and c is a homomorphism.Using this fact, next Proposition 3.2 expresses transformations of a holomorphicvolume form in a connected Calabi-Yau manifold in terms of a Lie algebra. Weassume H to be connected so that we express any h ∈ H as h = exp η · · · exp η l by η , · · · , η l ∈ h . For h , such η , · · · , η l ∈ h are not unique, however the followingholds for any of them. CONSTRUCTION OF SPECIAL LAGRANGIAN SUBMANIFOLDS 5
Proposition 3.2.
Let ( M, I, ω, Ω) be a connected Calabi-Yau manifold and H aconnected Lie group which acts on M preserving I and ω . Then there exists a H ∈ h ∗ such that for any h ∈ H , it holds that L ∗ h Ω = e √− (cid:104) a H ,η + ··· + η l (cid:105) Ω , where η , · · · , η l ∈ h such that h = exp η · · · exp η l . Proof.
Because c : H → U (1) defined above is a homomorphism, the followingcommutative relation holds between c and ( dc ) e : h ∼ = T e H → u (1): c ◦ exp ξ = e ( dc ) e ξ . In fact, since c makes a one-parameter subgroup exp( tξ ) of H into a one-parametersubgroup c (exp( tξ )) of U (1), there exists √− α ∈ u (1) ( α ∈ R ) such that c (exp( tξ )) = exp U (1) ( t ( √− α )) = e √− tα . By differentiating both sides, we ob-tain √− α = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 e √− tα = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 c (exp( tξ )) = ( dc ) e ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 exp( tξ ) = ( dc ) e ξ. Thus we see √− α = ( dc ) e ξ and ( c ◦ exp)( tξ ) = e t ( dc ) e ξ . When t = 1, we obtain c ◦ exp( ξ ) = e ( dc ) e ξ .Because H is connected, for each h ∈ H , there exist finite η , · · · , η l ∈ h suchthat h = exp η · · · exp η l . Then, we have c h = c (exp η · · · exp η l ) = c (exp η ) · · · c (exp η l ) = e ( dc ) e η · · · e ( dc ) e η l = e √− (cid:104)−√− dc ) e ,η (cid:105) · · · e √− (cid:104)−√− dc ) e ,η l (cid:105) = e √− (cid:104)−√− dc ) e ,η + ··· + η l (cid:105) . Therefore noting u (1) = {√− ϕ ∈ C | ϕ ∈ R } and letting a H := −√− dc ) e , wecan define a linear map a H : h → R , i.e., a H ∈ h ∗ and the claim of the propositionholds. (cid:3) By Proposition 3.2, transformations of a holomorphic volume form are expressedin terms of a Lie algebra. This enables us to explicitly show the Lagrangian angleof a Lagrangian immersion (
H/K ) × V → M which we construct in the next sectionin terms of the Lie algebra h at each ( hK, p ) ∈ ( H/K ) × V . Here K is a closed Liesubgroup of H and V is a submanifold in M . Corollary 3.1.
Let ( M, I, ω, Ω , H ) be same as Proposition 3.2. Then a H = 0 ifand only if the H -action preserves Ω , namely it preserves the Calabi-Yau structure. Special Lagrangian construction
In this section we show a construction of special Lagrangian submanifolds by“(generalized) perpendicular symmetries”, using the formula (Proposition 3.2)which we proved in the previous section. We construct an isotropic immersion,especially a Lagrangian immersion in Proposition 4.2. We give a formula thatexpress the Lagrangian angle of this Lagrangian immersion in Theorem 4.1. Wefinally construct a special Lagrangian immersion in Corollary 4.1 by considering acondition to have constant Lagrangian angle.
AKIFUMI OCHIAI
Immersions.
First with the use of group actions, we construct an immersionwhich is fundamental for our constructions. This immersion has a form H · V fora submanifold V in M . When H is abelian, it might be natural to assume thatthe action is free. Otherwise, we may need to consider singular orbits. To controlthem, we add a condition that the isotropy subgroup H p at each point p ∈ V is aconstant K . Lemma 4.1 is one of important properties that these immersions have. Proposition 4.1.
Let M be a manifold and H a Lie group which acts on M . Let h be the Lie algebra of H and V a submanifold of M . Assume the followings:(Imm- H ) ξ p / ∈ T p V \{ } for any p ∈ V and any ξ ∈ h , and(Imm-istp) the isotropy subgroup at p is a constant K for any p ∈ V .Define a map φ : ( H/K ) × V → M by ( hK, p ) (cid:55)→ hp . Then φ is an immersion. Lemma 4.1.
Assume the conditions of Proposition 4.1. For any ( hK, p ) ∈ ( H/K ) × V , any ξ ∈ h , and any v ∈ T p V , the following holds: φ ∗ ( hK,p ) (cid:18) ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h exp( tξ ) K, v (cid:19) = ( L h ) ∗ p ( ξ p + v ) . Proof of Lemma 4.1.
By (Imm-istp), the map φ is well-defined.Fix an arbitrary point ( hK, p ) ∈ ( H/K ) × V . First we show the following:T hK ( H/K ) = (cid:40) ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h exp( tξ ) K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ ∈ h (cid:41) . For g ∈ H , define τ g by τ g : ( H/K ) → ( H/K ); hK (cid:55)→ ghK. The map τ g is an element of Diff( H/K ), here Diff(
H/K ) is the space of all diffeo-morphisms on
H/K . We have T K ( H/K ) = { ddt (cid:12)(cid:12) t =0 exp( tξ ) K | ξ ∈ h } . We alsohave ( τ h ) ∗ K ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 exp( tξ ) K = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h exp( tξ ) K. The linear map ( τ h ) ∗ K : T K ( H/K ) → T hK ( H/K ) is an isomorphism. Thereforethe claim above holds.Let γ ( t ) be a curve in V that satisfies γ (0) = p and γ (cid:48) (0) = v . Then we have( φ ∗ ) ( hK,p ) (cid:18) ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h exp( tξ ) K, (cid:19) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h exp( tξ ) p = ( L h ) ∗ p ξ p , ( φ ∗ ) ( hK,p ) (0 , v ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 hγ ( t ) = ( L h ) ∗ p v. Thus Lemma 4.1 has been proved. (cid:3)
Proof of Proposition 4.1.
To prove Proposition 4.1, it is sufficient to show that iffor any ξ ∈ h and v ∈ T p V , φ ∗ ( hK,p ) (cid:18) ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h exp( tξ ) K, v (cid:19) = ( L h ) ∗ p ( ξ p + v ) = 0 , then ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h exp( tξ ) K = 0 , v = 0 . CONSTRUCTION OF SPECIAL LAGRANGIAN SUBMANIFOLDS 7
Since ( L h ) ∗ p is an isomorphism, if ( L h ) ∗ p ( ξ p + v ) = 0 , then ξ p + v = 0. By (Imm- H ), a pair ( ξ p , v ) is linearly independent. Hence we have ξ p = 0 and v = 0 from ξ p + v = 0. If we define a map j by j : ( H/K ) → ( H · p ); hK (cid:55)→ hp, the map j is a diffeomorphism. With the isomorphism j ∗ hK : T hK ( H/K ) → T hp ( H · p ), we have ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h exp( tξ ) K (cid:55)→ ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h exp( tξ ) p = L h ∗ p ξ p . Thus ddt (cid:12)(cid:12) t =0 h exp( tξ ) K = 0 if and only if ξ p = 0. (cid:3) Isotropic immersions.
Next we introduce a symplectic structure to a mani-fold M , and show a condition for the immersions of Proposition 4.1 to be isotropicin Proposition 4.2. Proposition 4.2.
Let ( M, ω ) be a n -dimensional symplectic manifold and H aLie group which acts on M and has a moment map µ . Let h be the Lie algebra of H and c an element of h ∗ . Let V c be a submanifold of M that satisfies V c ⊂ µ − ( c ) .Assume (Imm- H ), (Imm-istp) , and the followings: ( Istp- V c ) V c is isotropic, and(Istp-cnt) c is an element of Z ( h ∗ ) , the center of h ∗ .Define a map φ c : ( H/K ) × V c → M by ( hK, p ) (cid:55)→ hp . Then φ c is an isotropicimmersion.In addition, if the following condition holds, φ c is a Lagrangian immersion:(Lag-dim) dim H/K + dim V c = n . Lemma 4.2.
Assume the settings of Proposition 4.2 except the conditions (Imm- H ) , (Imm-istp) , (Istp- V c ) , (Istp-cnt) , and (Lag-dim) . Then ω p ( ξ p , v ) = 0 for any p ∈ V c , any v ∈ T p V c , and any ξ ∈ h .Proof of Lemma 4.2. Noting ( dµ ) p v = 0, we have ω p ( ξ p , v ) = − d ( (cid:104) µ ( · ) , ξ (cid:105) ) p v = −(cid:104) ( dµ ) p v, ξ (cid:105) = 0 . Thus we have shown Lemma 4.2. (cid:3)
Proof of Proposition 4.2.
Since the map φ c is an immersion by Proposition 4.1, itis sufficient for proving Proposition 4.2 to show that φ ∗ c ω ≡ H/K ) × V c . Fortwo arbitrary elements ( ddt (cid:12)(cid:12) t =0 h exp( tξ i ) K, v i ) ∈ T hK ( H/K ) × T p V c ( i = 1 , φ ∗ c ω ) ( hK,p ) (cid:18) ( ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h exp( tξ ) K, v ) , ( ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h exp( tξ ) K, v ) (cid:19) = ω hp (cid:18) ( φ c ) ∗ ( hK,p ) ( ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h exp( tξ ) K, v ) , ( φ c ) ∗ ( hK,p ) ( ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h exp( tξ ) K, v ) (cid:19) = ω hp (cid:18) ( L h ) ∗ p { ( ξ ) p + v } , ( L h ) ∗ p { ( ξ ) p + v } (cid:19) = ω p (( ξ ) p + v , ( ξ ) p + v )= ω p (( ξ ) p , ( ξ ) p ) + ω p (( ξ ) p , v ) + ω p ( v , ( ξ ) p ) + ω p ( v , v ) . AKIFUMI OCHIAI
The first term is equal to zero by (Istp-cnt), the second and third terms are zeroby Lemma 4.2, and the forth term is zero by (Istp- V c ). Thus we see that φ is anisotropic immersion. In addition, if (Lag-dim) holds, this immersion is Lagrangianby the definition of Lagrangian submanifolds. (cid:3) Lagrangian angle and special Lagrangian construction.
We con-structed a Lagrangian immersion in Proposition 4.2. We show a condition for thisimmersion to be a special Lagrangian immersion by using the Lagrangian angle. InTheorem 4.1, with the use of a formula for transformations of holomorphic volumeforms (Proposition 3.2), we give explicitly the Lagrangian angle of a Lagrangianimmersion of Proposition 4.2.Lemma 4.3 is used for calculations of the Lagrangian angle.
Theorem 4.1.
Let ( M, g, I, ω, Ω) be a connected n -dimensional Calabi-Yau man-ifold and H a connected Lie group which acts on M preserving I and has a momentmap µ . Let h be the Lie algebra of H and L a Lagrangian submanifold of M thathas a local Lagrangian angle θ . Let c be an element of h ∗ and V c a submanifold of M that satisfies V c ⊂ µ − ( c ) ∩ L . Assume (Imm-istp) , (Istp-cnt) , (Lag-dim) , andthe following (LagAng- H ) :(LagAng- H ) For any p ∈ V c and any ξ ∈ h , the following (i) and (ii) hold:(i) ξ p ∈ T ⊥ p L ⊕ T p V c ,(ii) ξ p / ∈ T p V c \{ } .Define φ c : ( H/K ) × V c → M as in Proposition 4.2. Then locally the followingholds: ( φ ∗ c Ω) ( hK,p ) = ± e √− θ c vol φ ∗ c g , where θ c ( hK, p ) = (cid:104) a H , η + · · · + η l (cid:105) + θ ( p ) − π H/K ) ,η , · · · , η l ∈ h such that h = exp η · · · exp η l . Lemma 4.3.
Under the conditions of Theorem 4.1, for any p ∈ V c there exist ξ , · · · , ξ m ∈ h , v , · · · , v n − m , w , · · · , w m ∈ T p V c that satisfy the followings: (1) For any h ∈ H , (cid:18) · · · , ( ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h exp( tξ j ) K, w j ) , · · · , · · · (0 , v i ) , · · · (cid:19) ( i = 1 , · · · , n − m, j = 1 , · · · , m ) is an orthonormal basis in T ( hK,p ) (( H/K ) × V c ) with respect to φ ∗ c g , (2) ( ξ j ) p + w j ∈ T ⊥ p L for j = 1 , · · · , m , and (3) (cid:18) I p { ( ξ ) p + w } , · · · , I p { ( ξ m ) p + w m } , v , · · · , v n − m (cid:19) is an orthonormalbasis in T p L with respect to ι ∗ g .Here m = dim( H/K ) , and ι : L → M is the embedding. Remark 4.1.
In Theorem 4.1, we do not assume the conditions (Imm- H ) and(Istp- V c ) in Proposition 4.2 to make φ c a Lagrangian immersion. However underthe conditions of Theorem 4.1, they hold. CONSTRUCTION OF SPECIAL LAGRANGIAN SUBMANIFOLDS 9
From Theorem 4.1 we immediately obtain the following corollary. Constructionsof special Lagrangian submanifolds are directly based on this corollary.
Corollary 4.1.
Assume the conditions of Theorem 4.1. In addition, if θ ≡ const . on V c (e.g. L : a special Lagrangian submanifold) and a H = 0 , then φ c is a specialLagrangian immersion. Now we prove Lemma 4.3, Theorem 4.1, and Remark 4.1.
Proof of Lemma 4.3.
First we show Lemma 4.3 (1) and (2). By Lemma 4.1, wehave (cid:18) φ c ∗ ( hK,p ) ( ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h exp( tξ j ) K, w j ) , φ c ∗ ( hK,p ) (0 , v i ) (cid:19) = ( L h ∗ p { ( ξ j ) p + w j } , L h ∗ p v i ) . Since L h ∗ p is isometric, it is enough for showing (1) and (2) to verify that thereexist ξ j , v i , and w j ( i = 1 , · · · , n − m, j = 1 , · · · , m ) such that (( ξ j ) p + w j , v i ) isan orthonormal system of T p M and ( ξ j ) p + w j ∈ T ⊥ p L .Noting (Lag-dim), let ( v i ) ( i = 1 , · · · , n − m ) be an orthonormal basis ofT p V c with respect to the metric on V induced from g . By (LagAng- H ), itholds that T p ( H · p ) ∩ T p L = { } . Hence, noting (Lag-dim) again, we cantake η j ∈ h ( j = 1 , · · · , m ) such that (( η j ) p ) is a basis of T p ( H · p ) and(( η ) p , · · · , ( η m ) p , v , · · · , v n − m ) is linearly independent in T p M .By (LagAng- H ), there exist u j ∈ T ⊥ p L \{ } and z j ∈ T p V c ( j = 1 , · · · , m ) thatdecompose ( η j ) p into direct summations as follows:( η j ) p = u j + z j ( j = 1 , · · · , m ) . ( u j ) is linearly independent. In fact, if u is expressed by u = b u + · · · + b m u m for b j ∈ R such that t ( b , · · · , b m ) (cid:54) = , we have( η ) p − z = b (( η ) p − z ) + · · · + b m (( η m ) p − z m ) ⇔ ( η ) p − { b ( η ) p + · · · + b m ( η m ) p } = z − ( b z + · · · + b m z m ) . Because the left-hand side belongs to T p ( H · p ), there exists η ∈ h such that η p equals to the left-hand side. If η p (cid:54) = 0, then η p ∈ T p V c \{ } because the right-hand side belongs to T p V c . This is contrary to (LagAng- H ). On the other hand, if η p = 0, this is contrary to that ( η j ) p is linearly independent because the left-handside equals to 0. For j = 2 , · · · , m , we can verify the same assertion. Thus ( u j ) islinearly independent.Therefore, noting u j ∈ T ⊥ p L , there exists A ∈ GL ( m, R ) such that (˜ u · · · , ˜ u m ) =( u , · · · , u m ) A is an orthonormal system in T ⊥ p L . Because T ⊥ p L ⊥ T p V c , ( v i , ˜ u j ) isan orthonormal system in T p M .Thus, if we define ξ j ∈ h and w j ∈ T p V c by (( ξ ) p , · · · , ( ξ m ) p ) =(( η ) p , · · · , ( η m ) p ) A and ( w , · · · , w m ) = ( − z , · · · , − z m ) A , then ˜ u j = ( ξ j ) p + w j and Lemma 4.3 (1) and (2) hold.Next we show Lemma 4.3 (3). By (1) and (2), it is enough for showing the claimof (3) to verify I p { ( ξ j ) p + w j } ∈ T ⊥ p V c ( j = 1 , · · · , m ). I p ( ξ j ) p ∈ T ⊥ p V c because 0 = ω p (( ξ j ) p , v i ) = g p ( I p ( ξ j ) p , v i ) by Lemma 4.2.On the other hand, I p w j ∈ T ⊥ p V c because V c is isotropic and 0 = ω p ( w j , v i ) = g p ( I p w j , v i ). Thus Lemma 4.3 (3) has been verified. (cid:3) Proof of Theorem 4.1.
Let X (0 , ( M ) be the set of complex vector fields of type(0 ,
1) on M . For any η ∈ h , it holds that η + √− Iη ∈ X (0 , ( M ). Since Ω is acomplex differential form of type ( n,
0) on M , we have i ( η )Ω = −√− i ( Iη )Ω . Take ( ξ j , v i , w j ) in Lemma 4.3 for i = 1 , · · · , n − m and j = 1 , · · · , m . Then,noting ( L ∗ h Ω) = e √− (cid:104) a H ,η + ··· + η l (cid:105) Ω, we have ( φ ∗ c Ω) ( hK,p ) (cid:18) · · · , ( ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h exp( tξ j ) K, w j ) , · · · , · · · , (0 , v i ) , · · · (cid:19) =Ω hp ( · · · , ( L h ) ∗ p { ( ξ j ) p + w j } , · · · , · · · , ( L h ) ∗ p v i , · · · )=( L ∗ h Ω) p ( · · · , ( ξ j ) p + w j , · · · , · · · , v i , · · · )=( −√− m ( L ∗ h Ω) p ( · · · , I p { ( ξ j ) p + w j } , · · · , · · · , v i , · · · )=( −√− m e √− (cid:104) a H ,η + ··· + η l (cid:105) Ω p ( · · · , I p { ( ξ j ) p + w j } , · · · , · · · , v i , · · · )=( −√− m e √− (cid:104) a H ,η + ··· + η l (cid:105) ( ι ∗ Ω) p ( · · · , I p { ( ξ j ) p + w j } , · · · , · · · , v i , · · · )=( −√− m e √− (cid:104) a H ,η + ··· + η l (cid:105) e √− θ (vol ι ∗ g ) p ( · · · , I p { ( ξ j ) p + w j } , · · · , · · · , v i , · · · )= ± e √− (cid:104) a H ,η + ··· + η l (cid:105) + θ − π m ) . By Lemma 4.3 (1), we havevol φ ∗ c g (cid:18) · · · , ( ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h exp( tξ j ) K, w j ) , · · · , · · · , (0 , v i ) , · · · (cid:19) = ± . Thus Theorem 4.1 has been proved. (cid:3)
Proof of Remark 4.1. (Imm- H ) holds by (LagAng- H ). Because L is a Lagrangiansubmanifold and V c ⊂ L , (Istp- V c ) holds. Thus we see that Remark 4.1 holds. (cid:3) Joyce pointed out in [11] that the commutativity of a Lie group is a necessarycondition for the group action to be perpendicular to whole of L . However, toconstruct a special Lagrangian submanifold, we need for a group action to be per-pendicular only to V c ⊂ L . Similarly the condition that L is a special Lagrangiansubmanifold, that is, the condition that the Lagrangian angle is constant on L isreduced to on V c . The perpendicular condition is also weakened as above. Thissituation, roughly speaking, indicates that a special Lagrangian submanifold maybe constructed by Corollary 4.1, if H · V c (not necessarily each fundamental vector ξ p at p ∈ V c ) is perpendicular to L for some c ∈ Z ( h ∗ ).As a special case of the condition (LagAng- H ) if we assume that each funda-mental vector ξ p is perpendicular to L , we obtain the next corollary. In this casewe need not assume (Istp-cnt) (see Remark 4.2). Corollary 4.2.
Let ( M, g, I, ω, Ω) be a connected n -dimensional Calabi-Yau man-ifold and H a connected Lie group which acts on M preserving I and has a momentmap µ . Let h be the Lie algebra of H and L a Lagrangian submanifold of M witha local Lagrangian angle θ . Let c be an element of h ∗ and V c a submanifold of M such that V c ⊂ µ − ( c ) ∩ L . Assume (Imm-istp) , (Lag-dim) , and (LagAng- H ) (cid:48) asfollows:(LagAng- H ) (cid:48) ξ p ⊥ T p L for any p ∈ V c , and any ξ ∈ h . CONSTRUCTION OF SPECIAL LAGRANGIAN SUBMANIFOLDS 11
Define φ c : ( H/K ) × V c → M as in Proposition 4.2. Then locally the followingholds: ( φ ∗ c Ω) ( hK,p ) = ± e √− θ c vol φ ∗ c g , where θ c ( hK, p ) = (cid:104) a H , η + · · · + η l (cid:105) + θ ( p ) − π H/K ) ,η , · · · , η l ∈ h such that h = exp η · · · exp η l . Remark 4.2.
Under the conditions of Corollary 4.2, (Imm- H ), (Istp- V c ), and(Istp-cnt) hold.Proof of Remark 4.2. (Imm- H ) holds by (LagAng- H ) (cid:48) . (Istp- V c ) holds as in Re-mark 4.1. Finally to show (Istp-cnt), we fix an arbitrary point hp ∈ H · p ( h ∈ H ).We have T hp ( H · p ) = (cid:26) ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h exp( tξ ) p (cid:12)(cid:12)(cid:12)(cid:12) ξ ∈ h (cid:27) = { ( L h ) ∗ p ξ p | ξ ∈ h } . Noting I p ξ p ∈ T p L and η p ∈ T ⊥ p L because of the assumption that L is Lagrangianand (LagAng- H ) (cid:48) , we have ω hp (( L h ) ∗ p ξ p , ( L h ) ∗ p η p ) = ( L ∗ h ω ) p ( ξ p , η p ) = g p ( I p ξ p , η p ) = 0 . Therefore H · p is isotropic. This is equivalent to µ ( p ) ∈ Z ( h ∗ ). (cid:3) Corollary 4.3.
Assume the conditions of Corollary 4.2. In addition, if θ ≡ const . on V c (e.g. L : a special Lagrangian submanifold) and a H = 0 , then φ c is a specialLagrangian immersion. Examples in T ∗ S n In this section, with the use of the results above, we construct non-trivial ex-amples of special Lagrangian submanifolds in non-flat Calabi-Yau manifolds T ∗ S n which equipped with the Stenzel metrics. In Subsection 5.1, we review some no-tions about the Stenzel metrics on T ∗ S n , and make sure some facts that is used toconstruct our examples. In Subsection 5.2, we construct two examples by using theactions of an abelian group. One of them is based on Corollary 4.1 of generalizedperpendicular conditions. In Subsection 5.3, we construct an example based onCorollary 4.3 by using the actions of a non-abelian group.Through this section, we use some notations. We denote e i by the column k -vector whose i -th element equals to 1 and the any other element equals to 0 in k -dimensional real or complex Euclidean space for some k ∈ N . We also denote ξ ij by ξ ij = E ji − E ij ∈ M ( k, R ) , here E ij denotes the k × k -matrix whose ( i, j )-component is 1 and all the othersare 0 for some k ∈ N .5.1. Stenzel metric on T ∗ S n . In [15], Stenzel constructed complete Ricci-flatK¨ahler metrics on the cotangent bundles of compact rank one symmetric spaces.This gives us examples of non-flat Calabi-Yau manifolds. In this paper, we denotethis Calabi-Yau structure by (T ∗ S n , I, ω Stz , Ω Stz ). We construct our examples ofspecial Lagrangian submanifolds in (T ∗ S n , I, ω Stz , Ω Stz ). We identify the tangent bundle and the cotangent bundle of the n -sphere S n ,and describe it byT ∗ S n = { ( x, ξ ) ∈ R n +1 × R n +1 | (cid:107) x (cid:107) = 1 , x (cid:5) ξ = 0 } , where “ (cid:5) ” is the canonical real inner product on the Euclidean space R n +1 and (cid:107) x (cid:107) = √ x (cid:5) x for each x ∈ R n +1 . We occasionally denote t ( x , · · · , x n +1 ) , t ( ξ , · · · , ξ n +1 ) by x, ξ respectively. SO ( n + 1) acts on T ∗ S n by h · ( x, ξ ) = ( hx, hξ ) for h ∈ SO ( n + 1) with cohomogeneity one. The principal orbitat a point ( x, ξ ) equals to a sphere bundle with a radius of (cid:107) ξ (cid:107) = √ ξ (cid:5) ξ .Let Q n be a complex quadric hypersurface in C n +1 as follows: Q n = (cid:40) z = t ( z , · · · , z n +1 ) ∈ C n +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n +1 (cid:88) i =1 z i = 1 (cid:41) . Sz¨oke gave an SO ( n + 1)-equivariant diffeomorphism from T ∗ S n to Q n in [16] asfollows: Φ : T ∗ S n → Q n ∈ ∈ ( x, ξ ) (cid:55)→ cosh( (cid:107) ξ (cid:107) ) x + √− (cid:107) ξ (cid:107) ) (cid:107) ξ (cid:107) ξ. We can induce a complex structure to Q n from C n +1 . Stenzel constructed Ricci-flat K¨ahler metrics with respect to these complex structures. We denoted this by I above. Therefore when we use the complex structure for studying the perpendicularconditions later, we do the calculations not in T ∗ S n but in Q n . The K¨ahler form ω Stz that Stenzel constructed is given as follows: ω Stz = √− ∂ ¯ ∂u ( r ) = √− n +1 (cid:88) i,j =1 ∂ ∂z i ∂ ¯ z j u ( r ) dz i ∧ d ¯ z j , here r = (cid:107) z (cid:107) = (cid:80) n +1 i =1 z i ¯ z i and u is a smooth real function satisfies the followingordinary differential equation:(1) ddt ( U (cid:48) ( t )) n = cn (sinh t ) n − ( c = const . > , here U ( t ) = u (cosh t ). The functions U and u has properties that U (cid:48) ( t ) > U (cid:48)(cid:48) ( t ) >
0, and u (cid:48) ( t ) > t > SO ( n + 1) preserve the Calabi-Yau structure of(T ∗ S n , I, ω Stz , Ω Stz ). Hence, for a H = a SO ( n +1) ∈ h ∗ = so ( n + 1) ∗ of Proposition3.2 determined by (T ∗ S n , I, ω Stz , Ω Stz , SO ( n + 1)), we have a H = 0 by Corollary3.1.A moment map µ : Q n → so ( n + 1) ∗ with respect to (T ∗ S n , ω Stz , SO ( n + 1)) isgiven in [1] as follows:(2) ( µ ( z ))( X ) = u (cid:48) ( r ) Iz (cid:5) Xz, ( z ∈ Q n , X ∈ so ( n + 1)) , here “ (cid:5) ” denotes the canonical real inner product on C n +1 .Finally, we give a basic fact for preparing an original special Lagrangian sub-manifold to construct a new one: Karigiannis and Min-Oo showed in [13] that aconormal bundle T ∗⊥ N in T ∗ S n for a submanifold N in S n is a special Lagrangiansubmanifold if and only if N is an austere submanifold of S n . Especially, a totallygeodesic submanifold of S n is an austere submanifold. CONSTRUCTION OF SPECIAL LAGRANGIAN SUBMANIFOLDS 13
The case of H = U (1) , L = T ∗⊥ S , L = T ∗⊥ S ⊂ T ∗ S . Let M be thecotangent bundle of 5-sphere T ∗ S , L the conormal bundle of a totally geodesicsubmanifold S of S , and L the conormal bundle of a totally geodesic submanifold S of S as follows: L ( ∼ = T ∗⊥ S ) = ( x x x , ξ ξ ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:107) x (cid:107) = 1 , ξ j ∈ R ( j = 2 , , ,L ( ∼ = T ∗⊥ S ) = ( x x , ξ ξ ξ ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:107) x (cid:107) = 1 , ξ j ∈ R ( j = 2 , , , . Because these S and S are totally geodesic submanifolds of S , they are aus-tere submanifolds. Hence their conormal bundles T ∗⊥ S and T ∗⊥ S are spe-cial Lagrangian submanifolds of T ∗ S . We use the polar coordinates x =cos ϕ cos ϕ , x = cos ϕ sin ϕ , x = sin ϕ for L and x = cos ϕ, x = sin ϕ for L .Let H be the U (1)-action of the Hopf-fibration S → C P , that is, the diagonal U (1) ∼ = SO (2)-action represented as follows: H ( ∼ = SO (2)) = h h h ∈ GL (6 , R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ∈ SO (2) . The Lie algebra h is given as follows: h ( ∼ = so (2)) = span { η } , here η = ξ + ξ + ξ and ξ ij is as mentioned at the beginning of this section.Note that the isotropy subgroup of this SO (2)-action is trivial at any point p ∈ L and L . Hence the condition (Imm-istp) holds for any point p ∈ L and L .We obtain an explicit expression of the moment map (2) by direct calculations. Lemma 5.1.
Define µ η by µ η ( z ) = (cid:104) µ ( z ) , η (cid:105) for z ∈ Φ( L j ) ( j = 1 , . Then µ η ( z ) equals to (cid:26) −K ( (cid:107) ξ (cid:107) )(cos ϕ cos ϕ ξ + cos ϕ sin ϕ ξ + sin ϕ ξ ) on Φ( L ) \{(cid:107) ξ (cid:107) = 0 } , −K ( (cid:107) ξ (cid:107) )(cos ϕξ + sin ϕξ ) on Φ( L ) \{(cid:107) ξ (cid:107) = 0 } . Here, K ( (cid:107) ξ (cid:107) ) = u (cid:48) (cosh(2 (cid:107) ξ (cid:107) )) sinh(2 (cid:107) ξ (cid:107) ) (cid:107) ξ (cid:107) , and u is the function defined by U ( t ) = u (cosh t ) for U which is a solution ofthe ordinary differential equation (1), that gives the K¨ahler potential of the Stenzelmetric. Under these preparations, we obtain the following:
Proposition 5.1.
Let ( M, I, ω
Stz , Ω Stz , L j , H ) be as above. Let V ( j ) c := L j ∩ µ − ( c ) for each c ∈ h ∗ and j = 1 , . (1) H · V (1) c is a special Lagrangian submanifold for any c ∈ h ∗ such that V (1) c (cid:54) = ∅ . (2) H · V (2) c is a special Lagrangian submanifold for any c ∈ h ∗ such that V (2) c (cid:54) = ∅ .Proof. Note that Z ( h ∗ ) = h ∗ because H is abelian. Thus (Istp-cnt) automaticallyholds. The condition (Imm-istp) also holds as mentioned above. Therefore we seethat (Istp-cnt) and (Imm-istp) hold in the any case of (1) and (2).(1) First we show the proposition above for j = 1. This proof is based onCorollary 4.3. We will show in order (1-I) the perpendicular condition: the H -action satisfies (LagAng- H ) (cid:48) on L , and (1-II) the submanifold condition: V (1) c (cid:54) = ∅ is a submanifold of M and (Lag-dim) holds for ( V (1) c , H, K ).(1-I) First we assume (cid:107) ξ (cid:107) (cid:54) = 0. By direct calculations, for z ∈ Φ( L ), thefundamental vector η z and I z η z are given as follows: η z = cosh( (cid:107) ξ (cid:107) ) ϕ cos ϕ ϕ sin ϕ ϕ + √− (cid:107) ξ (cid:107) ) (cid:107) ξ (cid:107) − ξ − ξ − ξ ,I z η z = sinh( (cid:107) ξ (cid:107) ) (cid:107) ξ (cid:107) ξ ξ ξ + √− (cid:107) ξ (cid:107) ) ϕ cos ϕ ϕ sin ϕ ϕ . On the other hand, using the coordinates above, we have a basis of T z Φ( L ) asfollows: ∂∂ϕ = cosh( (cid:107) ξ (cid:107) )( − sin ϕ cos ϕ e − sin ϕ sin ϕ e + cos ϕ e ) ,∂∂ϕ = cosh( (cid:107) ξ (cid:107) )( − cos ϕ sin ϕ e + cos ϕ cos ϕ e ) ,∂∂ξ j = sinh( (cid:107) ξ (cid:107) ) (cid:107) ξ (cid:107) ξ j x + √− (cid:26) ξ j (cid:107) ξ (cid:107) F ξ + sinh( (cid:107) ξ (cid:107) ) (cid:107) ξ (cid:107) e j (cid:27) ( j = 2 , , , where F = F ( (cid:107) ξ (cid:107) ) = cosh( (cid:107) ξ (cid:107) ) − sinh( (cid:107) ξ (cid:107) ) (cid:107) ξ (cid:107) . Since L is a Lagrangian submanifold of a K¨ahler manifold M , it is sufficient forverifying η z ∈ T ⊥ z Φ( L ) to show I z η z ∈ T z Φ( L ). For generating the imagi-nary part of I z η z by (cid:16) ∂∂ξ j (cid:17) ( j = 2 , , CONSTRUCTION OF SPECIAL LAGRANGIAN SUBMANIFOLDS 15 ( a , a , a ) ∈ R \{ } which satisfies(3) A a a a = cosh( (cid:107) ξ (cid:107) ) cos ϕ cos ϕ cosh( (cid:107) ξ (cid:107) ) cos ϕ sin ϕ cosh( (cid:107) ξ (cid:107) ) sin ϕ , where A = ξ (cid:107) ξ (cid:107) F + sinh( (cid:107) ξ (cid:107) ) (cid:107) ξ (cid:107) ξ ξ (cid:107) ξ (cid:107) F ξ ξ (cid:107) ξ (cid:107) F ξ ξ (cid:107) ξ (cid:107) F ξ (cid:107) ξ (cid:107) F + sinh( (cid:107) ξ (cid:107) ) (cid:107) ξ (cid:107) ξ ξ (cid:107) ξ (cid:107) F ξ ξ (cid:107) ξ (cid:107) F ξ ξ (cid:107) ξ (cid:107) F ξ (cid:107) ξ (cid:107) F + sinh( (cid:107) ξ (cid:107) ) (cid:107) ξ (cid:107) . With the use of series expansion of hyperbolic functions, we can see rank A = 3 if (cid:107) ξ (cid:107) (cid:54) = 0. Hence (3) has a non-trivial solution for each (cid:107) ξ (cid:107) (cid:54) = 0. For this solution( a , a , a ), we can verify that there exists ( b , b ) ∈ R which satisfies b ∂∂ϕ + b ∂∂ϕ + a ∂∂ξ + a ∂∂ξ + a ∂∂ξ = I z η z by using the relation (3). Thus if (cid:107) ξ (cid:107) (cid:54) = 0, (LagAng- H ) (cid:48) holds.When (cid:107) ξ (cid:107) = 0, by taking a limit (cid:107) ξ (cid:107) →
0, we have ∂∂ξ j → √− e j ( j = 2 , , . Thus we can also verify that I z η z ∈ T z Φ( L ) when (cid:107) ξ (cid:107) = 0.(1-II) Note that µ ( L ∩ {(cid:107) ξ (cid:107) = 0 } ) = 0. When (cid:107) ξ (cid:107) (cid:54) = 0, we use the followingfact: There exists a neighborhood U p around p ∈ V (1) c in L such that V (1) c ∩ U p isa submanifold of L (therefore of M ), if ( ∇ µ η ) p (cid:54) = ∈ T p L , and then dim V (1) c =dim L − − ∇ is the gradient with respect to the induced metric ι ∗ g Stz by the inclusion map ι : L → M . Using the properties of Stenzel’s K¨ahlerpotential u , we can verify that ∇ µ η (cid:54) = on L \{(cid:107) ξ (cid:107) = 0 } by direct calculations.When (cid:107) ξ (cid:107) = 0, it is sufficient to verify V (1)0 is a submanifold of M . By the expressionof the moment map in Lemma 5.1, we have V (1)0 = ( x x x , ξ ξ ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:107) x (cid:107) = 1 , x x x (cid:5) ξ ξ ξ = 0 . This is diffeomorphic to T S . Therefore V (1) c (cid:54) = ∅ is a submanifold of M for any c ∈ h ∗ , and (Lag-dim) holds for ( V (1) c , H, K ). Thus we have proven (1) of theproposition.(2) This proof is based on Corollary 4.1. (2-I) the generalized perpendicularcondition: the H -action satisfies (LagAng- H ) on L . To show it, first we assume (cid:107) ξ (cid:107) (cid:54) = 0. By direct calculations, we have η z = cosh( (cid:107) ξ (cid:107) ) ϕ ϕ + √− (cid:107) ξ (cid:107) ) (cid:107) ξ (cid:107) − ξ − ξ − ξ ξ We set the following strategy. First we decompose I z η z as follows: I z η z = I z ( η z ) + I z ( η z ) , here I z ( η z ) = sinh( (cid:107) ξ (cid:107) ) (cid:107) ξ (cid:107) ξ ξ + √− (cid:107) ξ (cid:107) ) ϕ ϕ , I z ( η z ) = sinh( (cid:107) ξ (cid:107) ) (cid:107) ξ (cid:107) ξ − ξ . Then assume that I z ( η z ) ∈ T z Φ( L ). Since I z ( η z ) clearly has no T z Φ( L )-components, we see that the decomposition above is a direct decomposition withrespect to T z Q n = T z Φ( L ) ⊕ T ⊥ z Φ( L ). Since( η z ) = √− (cid:107) ξ (cid:107) ) (cid:107) ξ (cid:107) − ξ ξ and µ depend neither on fifth nor sixth component of the imaginary part of C n +1 ∼ =T z C n +1 ⊃ T z Q n , we have (cid:104) ( dµ η ) z , ( η z ) (cid:105) = 0 , namely ( η z ) ∈ T z µ − ( µ ( z )). Hence we have that ( η z ) ∈ T z µ − ( µ ( z )) ∩ T z Φ( L ) = T z Φ( V (2) µ ( z ) ). Noting that ( η z ) (cid:54) = for any z ∈ Φ( L ), we thus see that(LagAng- H ) holds if I z ( η z ) ∈ T z Φ( L ). We can verify I z ( η z ) ∈ T z Φ( L ) actu-ally as in the case of (1). Hence (LagAng- H ) holds at any point p ∈ L \{(cid:107) ξ (cid:107) = 0 } .When (cid:107) ξ (cid:107) = 0, we can also compute I z η z ∈ T z Φ( L ) by taking a limit (cid:107) ξ (cid:107) → H ) (cid:48) holds at any point p ∈ L ∩ {(cid:107) ξ (cid:107) = 0 } morestrictly. Thus we see that (2-I) holds.(2-II) the submanifold condition: V (2) c (cid:54) = ∅ is a submanifold of M and (Lag-dim) holds for ( V (2) c , H, K ). For c (cid:54) = 0, we can verify this as in the case of (1) byconsidering the gradient ∇ µ η . For c = 0, V (2)0 is expressed as follows: V (2)0 = ( x x , ξ ξ ξ ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:107) x (cid:107) = 1 , (cid:20) x x (cid:21) (cid:5) (cid:20) ξ ξ (cid:21) = 0 . CONSTRUCTION OF SPECIAL LAGRANGIAN SUBMANIFOLDS 17
Ignoring fifth and sixth components, this is diffeomorphic to
T S ∼ = S × R . Fifthand sixth components constitute a plane unrelated to the base manifold. Thus wesee V (2)0 ∼ = S × R and ( V (2)0 , H, K ) satisfies (Lag-dim). Thus we see that (2-II)holds. (cid:3) The case of H = SO (2) × SO (2) × SO (3) , L = T ∗⊥ S ⊂ T ∗ S . Let M be thecotangent bundle of 6-sphere T ∗ S and L the conormal bundle of a totally geodesicsubmanifold S of S as follows: L (= T ∗⊥ S ) = ( x x x , ξ ξ ξ ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:107) x (cid:107) = 1 , ξ j ∈ R ( j = 2 , , , . Define H as follows: H ( ∼ = SO (2) × SO (2) × SO (3))= h h
00 0 h ∈ GL (7 , R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h , h ∈ SO (2) , h ∈ SO (3) . Note that H is non-abelian. The Lie algebra h of H is given as follows: h ( ∼ = so (2) ⊕ so (2) ⊕ so (3)) = span { ξ , ξ , ξ , ξ , ξ } , here ξ ij is as mentioned at the beginning of this section.We obtain an explicit expression of the moment map of (2) by direct calcu-lations. Define µ ij for the basis ( ξ , ξ , ξ , ξ , ξ ) of h and z ∈ Φ( L ) by µ ij ( z ) = (cid:104) µ ( z ) , ξ ij (cid:105) . Lemma 5.2.
For ( M, I, ω, H, µ ) above, and z ∈ Φ( L ) \{(cid:107) ξ (cid:107) = 0 } , we have µ ( z ) = −K ( (cid:107) ξ (cid:107) ) cos ϕ cos ϕ ξ ,µ ( z ) = −K ( (cid:107) ξ (cid:107) ) cos ϕ sin ϕ ξ ,µ ( z ) = −K ( (cid:107) ξ (cid:107) ) sin ϕ ξ ,µ ( z ) = −K ( (cid:107) ξ (cid:107) ) sin ϕ ξ ,µ ( z ) ≡ . Here, we use the polar coordinates x = cos ϕ cos ϕ , x = cos ϕ sin ϕ , x =sin ϕ , and K ( (cid:107) ξ (cid:107) ) = u (cid:48) (cosh(2 (cid:107) ξ (cid:107) )) sinh(2 (cid:107) ξ (cid:107) ) (cid:107) ξ (cid:107) . We obtain the following result:
Proposition 5.2.
Let ( M, I, ω
Stz , Ω Stz , L, H ) be as above. Define a rank two sub-bundle ˆ L of L as follows: ˆ L = ( x x x , ξ ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:107) x (cid:107) = 1 , ξ j ∈ R ( j = 2 , . Let ˆ L pr be the set of all points p ∈ ˆ L such that the isotropy subgroup H p satisfies H p ⊂ H q for all q ∈ ˆ L . For ( c , c ) ∈ R , define V ( c ,c ) and ˆ V ( c ,c ) by V ( c ,c ) = ˆ L pr ∩ { p ∈ M | µ ( p ) = c , µ ( p ) = c , µ ij ( p ) = 0 } , ˆ V ( c ,c ) = ˆ L ∩ { p ∈ M | µ ( p ) = c , µ ( p ) = c , µ ij ( p ) = 0 } , here ( i, j ) = (5 , , (5 , , (6 , . Then for any ( c , c ) (cid:54) = (0 , ∈ R such that V ( c ,c ) (cid:54) = ∅ , H · V ( c ,c ) is a special Lagrangian submanifold of M , and H · ˆ V (0 , isa union of five connected special Lagrangian submanifolds of M .Proof. The proof for V ( c ,c ) is based on Corollary 4.3, and one for ˆ V (0 , on directcalculations. As we saw in Remark 4.2, V ( c ,c ) has to be included in the inverseimage of the center Z ( h ∗ ) of h ∗ with the moment map µ . Hence, noting the so (3)-part of h ∼ = so (2) ⊕ so (2) ⊕ so (3), we see that we can apply our construction forthe part such that µ ij ( p ) = 0 (( i, j ) = (5 , , (5 , , (6 , L . This indicates that ξ = ξ = 0 is necessary. That is, the place in where we have to check the conditionsof Corollary 4.2 is ˆ L ⊂ L .By the definition of V ( c ,c ) , at any point p ∈ V ( c ,c ) , the isotropy subgroup H p is the following one-parameter subgroup K generated by ξ : K ( ∼ = SO (2)) = (cid:26) (cid:20) E h (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) h ∈ SO (2) (cid:27) , here E is the unit 5 × V ( c ,c ) for ( c , c ) (cid:54) = (0 , µ ( ˆ L ∩{(cid:107) ξ (cid:107) =0 } ) = , we can assume (cid:107) ξ (cid:107) (cid:54) = 0 in this case. As same as Proposition 5.1, con-ditions we have to check are the followings: (I) the perpendicular condition: the H -action satisfies (LagAng- H ) (cid:48) on ˆ L \{(cid:107) ξ (cid:107) = 0 } , and (II) the submanifold condi-tion: V ( c ,c ) (cid:54) = ∅ is a submanifold of M and (Lag-dim) holds for ( V ( c ,c ) , H, K ).We can verify these in the same way as Proposition 5.1.Finally we study ˆ V (0 , generally rather than V (0 , , including non-principalpoints. By Lemma 5.2, We obtain thatˆ V (0 , = ( x x x , ξ ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:107) x (cid:107) = 1 ,ξ , ξ ∈ R ,x ξ = x ξ = 0 . CONSTRUCTION OF SPECIAL LAGRANGIAN SUBMANIFOLDS 19 ˆ V (0 , is not a smooth manifold. However it is a union, which is not disjoint, of thefollowing five connected manifolds:ˆ V (0 , = ˆ V S (0 , ∪ ˆ V S × R (0 , , (1) ∪ ˆ V S × R (0 , , (3) ∪ ˆ V R (0 , , (1) ∪ ˆ V R (0 , , ( − , hereˆ V S (0 , = ( x x x , ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:107) x (cid:107) = 1 , ˆ V S × R (0 , , (1) = ( x x , ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:107) x (cid:107) = 1 ,ξ ∈ R , ˆ V S × R (0 , , (3) = ( x x , ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:107) x (cid:107) = 1 ,ξ ∈ R , ˆ V R (0 , , ( (cid:15) ) = ( (cid:15) , ξ ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ , ξ ∈ R , and (cid:15) = ±
1. We can see that each set ˆ V W (0 , is a 2-dimensional connected submani-fold of M diffeomorphic to W . Each ˆ V W (0 , has non-principal orbits with respect tothe action of H on H · ˆ V W (0 , . Hence it does not satisfy (Imm-istp). However we candirectly verify that each H · ˆ V W (0 , for ˆ V S (0 , , ˆ V S × R (0 , , ( j ) ( j = 1 , V R (0 , , ( (cid:15) ) ( (cid:15) = ± M diffeomorphic to S , T ∗⊥ S , and T ∗⊥ S respectively. (cid:3) We chose SO (2) × SO (2) × SO (3) as a Lie group H for the special Lagrangiansubmanifold L ⊂ T ∗ S of Proposition 5.2 rather than SO (2) × SO (5) because of tworeasons. First, since the center of the Lie algebra h ∼ = so (2) ⊕ so (2) ⊕ so (3) has twodimensions, we could obtain two-parameters of special Lagrangian submanifolds H · V ( c ,c ) . Second, for p, q ∈ N such that p + q = n + 1, special Lagrangiansubmanifolds which are SO ( p ) × SO ( q )-invariant in (T ∗ S n , I, g Stz , ω
Stz ) have beenalready obtained by Hashimoto and Sakai in [5], and they showed that such specialLagrangian submanifolds are cohomogeneity one with respect to SO ( p ) × SO ( q ).In the case of Proposition 5.2, we can verify that SO (2) × SO (2) × SO (3) acts on H · V ( c ,c ) with cohomogeneity two. References [1] H. Anciaux,
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