Rigidity theorems for minimal Lagrangian surfaces with Legendrian capillary boundary
aa r X i v : . [ m a t h . DG ] J u l Rigidity theorems for minimal Lagrangian surfaces with Legendrian capillaryboundary ✩ Yong Luo a,b , Linlin Sun a,b, ∗ a School of Mathematics and Statistics, Wuhan University, 430072 Wuhan, Hubei, China b Hubei Key Laboratory of Computational Science, Wuhan University, 430072 Wuhan, Hubei, China
Abstract
In this note, we study minimal Lagrangian surfaces in B with Legendrian capillary boundary on S . On the one hand,we prove that any minimal Lagrangian surface in B with Legendrian free boundary on S must be an equatorial planedisk. One the other hand, we show that any annulus type minimal Lagrangian surface in B with Legendrian capillaryboundary on S must be congruent to one of the Lagrangian catenoids. These results confirm the conjecture proposedby Li, Wang and Weng (Sci. China Math., 2020). Keywords:
Minimal Lagrangian surface, Legendrian capillary boundary, Lagrangian catenoid
1. Introduction
Let C n = R n be the standard complex plane with its canonical K¨ahler form ω and almost complex structure J . Let S n − be the (2 n − n -dimensinal submanifold Σ n in C n is called a Lagrangain submanifold if JT Σ n = T ⊥ Σ n , where T ⊥ Σ n denotes the normal space of Σ n in C n , andan ( n −
1) dimensional submanifold K n − in S n − is called a Legendrian submanifold if R ⊥ T K n − , where R is theReeb field of S n − with R ( x ) = Jx for every x ∈ S n − .It is well known that Lagrangian submanifolds in a complex space form have many similarities with hypersurfacesin a real space form. Recently, inspired by the study of capillary hypersurfaces M in B n + ⊂ R n + , which haveconstant mean curvature, non-empty boundary such that ˚ M ⊂ ˚ B n + and ∂ M ⊂ ∂ B n + = S n , which intersect ∂ B n + with a constant angle, Li, Wang and Weng [11] initiated the very interesting study of Lagrangian submanifolds withLegendrian capillary boundary in B n ⊂ C n .First let us recall some definitions introduced in [11]. Let x : Σ n → B n be a Lagrangian submanifold with ∂ Σ n ⊂ ∂ B n = S n − being a Legendrian submanifold. Li, Wang and Weng observed that the unit normal ν at x ∈ ∂ Σ n ⊂ Σ n lies in the plane spanned by x and Jx , i.e. there exists a θ ∈ [0 , π ) such that ν = sin θ x + cos θ Jx . The angle θ is called a contact angle and Σ n is called a Lagrangian submanifold with Legendrian capillary boundary(or simply capillary Lagrangian submanfold), if the contact angle is a local constant. When θ = π , Σ n is called aLagrangian submanifold with Legendrian free boundary, or a free boundary Lagrangian submanifold. ✩ After our proofs of Theorem 3.3 was completed, we learned from Professor Guofang Wang that he had also noticed that the boundary ofminimal Lagrangian surfaces in B with Legendrian capillary boundary on S are great circles and then he got an idea to prove Conjecture 1 whichcould be quite standard. The authors would like to thank him for the encouragement for us to complete and submit our paper. Many thanks to Dr.Qing Cui, Jiabin Yin and Jingyong Zhu for their interests in this paper and discussions. This research is partially supported by the National NaturalScience Foundation of China (Grant Nos. 11971358, 11801420) and the Youth Talent Training Program of Wuhan University. The second authoralso would like to express his gratitude to Professor J¨urgen Jost for his invitation to MPI MIS for their hospitality. ∗ Corresponding author.
Email addresses: [email protected] (Yong Luo), [email protected] (Linlin Sun)
Preprint submitted to XXX July 8, 2020 hen n =
2, typical examples of minimal Lagrangian surfaces in B with Legendrian capillary boundary are theequatorial plane disk and the Lagrangian catenoids, as discussed in [11] (see also Example 3.1). Note that the contactangle for the equatorial plane disk is π , but the contact angle for Lagrangian catenoids are constants which are notequal to π . Li, Wang and Weng [11] got the following Nitche (or Hopf) type rigidity theorem. Theorem 1.1 (Li, Wang and Weng) . Given D : = n ( x , x ) : x + x ≤ o . Let x : D −→ B be a (branched) minimalLagrangian surface with Legendrian capillary boundary on S . Then x ( D ) is an equatorial plane disk. This theorem is the Lagrangian counterpart of related results for capillary surfaces in B n by Nitsche [14], Ros andSouam [15] and Fraser and Schoen [5]. Then they conjectured that:There is no annulus type minimal Lagrangian surface with Legendrian free boundary.Moreover, they wrote down the following ([11, Conjecture 2.16]). Conjecture 1.
Any embedded annulus type minimal Lagrangian surface with Legendrian capillary boundary on S is one of the Lagrangian catenoids. This conjecture is the Lagrangian counterpart of the conjecture for free boundary minimal surfaces in B proposedby Fraser and Li [4]. Conjecture 2 (Fraser-Li) . The critical catenoid is the unique embedded free boundary minimal annulus in B . In this paper, we first show that Lagrangian minimal surfaces in B with Legendrian free boundary must be anequatorial plane disk (Theorem 3.1), which extends Theorem 1.1 in the Legendrian free boundary case and confirmsthe statement:There is no annulus type minimal Lagrangian surface with Legendrian free boundary.Finally, we give an a ffi rmative answer to Conjecture 1. Actually, we prove that Conjecture 1 is true without theembeddedness assumption (Theorem 3.3).As it is well known, hypersurfaces in a real space form have many similarities with Lagrangian submanifolds in acomplex space from, and many rigidity results for minimal hypersurfaces in a real space form have their Lagrangiancounterparts. But according to our knowledge, rigidity results in the Lagrangian submanifolds case are always muchmore complicated and their proofs (if they exist) need more job. Consequently, although some rigidity results are truefor minimal hypersurfaces in a real space form, their Lagrangian counterparts are still open. For example Brendle [1]proved the longstanding Lawson’s conjecture, which states that the Cli ff ord torus is the unique embedded minimal toriin S . But its Lagrangian counterpart, that is if embedded minimal Lagrangian tori in CP are given by the examplesconstructed by Haskins [7] with certain symmetry (see also [2, 8]), remains widely open. Another example is theconjecture given by the authors ([13, Conjecture 1]) of this paper on the first pinching constant of closed minimalLagrangian submanifolds in CP n , while the case of closed minimal hypersurfaces was established by Simons [12],Chern, do Carmo and Kobayashi [3] and Lawson [10].Bewaring of this, it would be a surprise for us to see that though Fraser and Li’s conjecture, i.e. Conjecture 2,remains open, but its Lagrangian counterpart, i.e. Conjecture 1, could be verified. The above mentioned Nitsche (orHopf) type rigidity results for capillary surfaces [5, 14, 15] and Theorem 1.1 were proved by the technique of Hopf’sholomorphic cubic form. While in our proof of Conjecture 1 we use simultaneously Hopf’s holomorphic cubicform and a maximum principle for surfaces with boundary, which is quite subtle. The main observation is that, theboundary of a minimal Lagrangian submanifold in B n with Legendrian capillary boundary on S n − is still minimal(see Lemma 2.2), which enable us to use the maximum principle. It would be very interesting to see if this methodis workable for Fraser and Li’s conjecture, by exploring more boundary properties of the critical catenoid. Here wewould like to point out that Li [12] observed that by a Bj¨orling-type uniqueness result for free boundary minimalsurfaces of Kapouleas and Li [9], to prove Fraser and Li’s conjecture, it su ffi ces to show that one of the boundarycomponents of the minimal annulus is rotationally symmetric. We invite the readers who desire more information onFraser and Li’s conjecture to consult the recent excellent surveys by Li [12] and Wang and Xia [16] and referencestherein. See also Fraser and Schoen [6] for a very deep characterization of the critical cateniod.The rest of this paper is organized as follows. In section 2 we give some properties of the Legendrian boundaryand contact angle. Main results of this paper and their proofs are given in section 3.2 . Properties of the Legendrian boundary and contact angle Let x : Σ n −→ B n be an immersed Lagrangian submanifold with boundary ∂ Σ n on the unit round sphere S n − .Let ν be the unit outward normal vector field of ∂ Σ n ֒ → Σ n . Since Σ n is a Lagrangian submanifold of B n , on theboundary we have the following orthogonal decomposition T B n | ∂ Σ n = T Σ n | ∂ Σ n ⊕ T ⊥ Σ n | ∂ Σ n = T Σ n | ∂ Σ n ⊕ JT Σ n | ∂ Σ n = T ∂ Σ n ⊕ JT ∂ Σ n ⊕ span { ν, J ν } . Notice that T B n | ∂ Σ n = T S n − | ∂ Σ n ⊕ span { x } = T ∂ Σ n ⊕ T ⊥ (cid:16) ∂ Σ n ֒ → S n − (cid:17) ⊕ span { x } . Therefore ∂ Σ n is a Legendrian submanifold of S n − if and only if T ⊥ (cid:16) ∂ Σ n ֒ → S n − (cid:17) = JT ∂ Σ n ⊕ span { Jx } , if and only if span { ν, J ν } = span { x , Jx } , which is equivalent to that ν = sin θ x + cos θ Jx , (2.1)where θ : ∂ Σ n −→ [0 , π ) is a smooth function. The angle θ is called a contact angle.Let B , B Σ and B ∂ be the second fundamental form of the isometric immersions Σ n ֒ → B n , ∂ Σ n ֒ → Σ n and ∂ Σ n ֒ → S n − respectively. Let H , H Σ and H ∂ be the mean curvature vector of the isometric immersions Σ n ֒ → B n , ∂ Σ n ֒ → Σ n and ∂ Σ n ֒ → S n − respectively. Finally, let ¯ ∇ , ∇ and ∇ ∂ be the Levi-Civita connections on B n , Σ n and ∂ Σ n respectively. Lemma 2.1.
For all X , Y , Z ∈ T ∂ Σ n , B Σ ( X , Y ) = − sin θ h X , Y i ν, (2.2) h B ( X , Y ) , J ν i = cos θ h X , Y i , (2.3) h B ( X , Y ) , JZ i = D B ∂ ( X , Y ) , JZ E . (2.4) Moreover, ∇ ∂ θ = J B ( ν, ν ) . (2.5) Proof.
On the one hand, the isometric immersions ∂ Σ n ֒ → Σ n ֒ → B n implies¯ ∇ X Y = ∇ ∂ X Y + B Σ ( X , Y ) + B ( X , Y ) . On the other hand, the isometric immersions ∂ Σ n ֒ → S n − ֒ → B n gives¯ ∇ X Y = ∇ ∂ X Y + B ∂ ( X , Y ) − h X , Y i x . Thus B Σ ( X , Y ) + B ( X , Y ) = B ∂ ( X , Y ) − h X , Y i x . B Σ ( X , Y ) = − sin θ h X , Y i ν, h B ( X , Y ) , J ν i = cos θ h X , Y i , h B ( X , Y ) , JZ i = D B ∂ ( X , Y ) , JZ E . Finally, a direct calculation yields h B ( X , ν ) , J ν i = D ¯ ∇ X ν, J ν E = h− X ( θ ) J ν + sin θ X + cos θ JX , J ν i = − X ( θ ) . Hence ∇ ∂ θ = J B ( ν, ν ) . (cid:3) Define η = ι H ω | Σ n , η ∂ = ι H ∂ ω | ∂ Σ n . The one forms η and η ∂ are called the Maslov form of the Lagrangian immersion Σ n ֒ → B n and the Legendrianimmersion ∂ Σ n ֒ → S n − respectively. Equality (2.3) implies that ι ν η = − h B ( ν, ν ) , J ν i − ( n −
1) cos θ. (2.6)Equalities (2.4) and (2.5) yield η | ∂ Σ n = η ∂ + d θ. (2.7)By (2.7) we obtain the following very important observation. Lemma 2.2. If Σ n is a minimal Lagrangian submanifold in B n with Legendrian capillary boundary, then ∂ Σ n is aminimal Legendrian submanifold in S n − .
3. Main results and proofs
In this section, we assume x : Σ −→ B is a minimal Lagrangian surface with Legendrian capillary boundary on S , i.e., the contact angle θ is a local constant.Then by Lemma 2.2 each component of ∂ Σ is a Legendrian geodesic curve and hence a Legendrian great circle in S .When restricted on ∂ Σ , we have from (2.2) and (2.6) that κ g = sin θ, B ( ν, ν ) = − cos θ J ν. (3.1)Here κ g is the geodesic curvature of the curve ∂ Σ in Σ . Let z be a local conformal coordinates on Σ and consider thecubic form Q on Σ defined by Q = h B ( ∂ z , ∂ z ) , J ∂ z i (d z ) . Since Σ is minimal, we know that Q is holomorphic. We have Theorem 3.1.
Let Σ be a minimal Lagrangian surface in B with Legendrian free boundary on S . Then Σ is anequatorial plane disk. roof. If Σ is Lagrangian submanifold with Legendrian free boundary, i.e., θ = π , when restricted on ∂ Σ , by (3.1) wehave κ g = , B ∂ = . Hence Q = ∂ Σ , which implies that Q = Σ . Consequently, Σ is totally geodesic in B .Applying the Gauss-Bonnet formula we have2 π (cid:2) − γ ) − r (cid:3) = πχ ( Σ ) = Z Σ κ + Z ∂ Σ κ g = Z ∂ Σ = π r , where κ is the Gauss curvature of Σ , γ is the genus of Σ and r the numbers of the components of ∂ Σ . Thus γ + r = . Consequently, γ = r =
1. Therefore Σ is a topological disk and is an equatorial plane disk according to Li,Wang and Weng’s result (Theorem 1.1). (cid:3) In particular, we have proved the following.
Corollary 3.2.
There is no minimal Lagrangian annulus in B with Legendrian free boundary on S . Next we will prove Conjecture 1 in the introduction. Before that, let us recall the example of Lagrangian catenoidsand give some detailed descriptions on them, which will be helpful to understand our proofs presented below.
Example 3.1 (Lagrangian catenoids) . We identify a real vector (cid:16) x , x , y , y (cid:17) ∈ R as a complex vector (cid:16) z , z (cid:17) = (cid:16) x + √− y , x + √− y (cid:17) ∈ C . The Lagrangian catenoid in R can be identified as the holomorphic curve Σ λ in C ,with respect to the standard K¨ahler form √− P k = d z k ∧ d¯ z k , given by Σ λ = (cid:26)(cid:18) z , λ z (cid:19) : z ∈ C \ { } (cid:27) , where λ ∈ R \ { } . Let Ω = d z ∧ d z be the holomorphic symplectic form on C . Then Ω | Σ λ = . Hence Σ λ is a Lagrangian surface in C with respect to the K¨ahler form Re Ω (or Im Ω ). Notice that the complexstructure J associated with the K¨ahler form Re Ω = d x ∧ d x − d y ∧ d y is J (cid:16) x , x , y , y (cid:17) = (cid:16) − x , x , y , − y (cid:17) . Let z = re √− φ where ( r , φ ) is the polar coordinates. Then Σ λ = (cid:26)(cid:18) r cos φ, λ r cos φ, r sin φ, − λ r sin φ (cid:19) : r > , ≤ φ < π. (cid:27) Set X ( r , φ ) = (cid:18) r cos φ, λ r cos φ, r sin φ, − λ r sin φ (cid:19) . The tangent bundle T Σ λ is spanned by X r = (cid:18) cos φ, − λ r cos φ, sin φ, λ r sin φ (cid:19) , X φ = (cid:18) − r sin φ, − λ r sin φ, r cos φ, − λ r cos φ (cid:19) , T ⊥ Σ λ is spanned by JX r = (cid:18) λ r cos φ, cos φ, λ r sin φ, − sin φ (cid:19) , JX φ = (cid:18) λ r sin φ, − r sin φ, − λ r cos φ, − r cos φ (cid:19) . One can check that | X r | − r (cid:12)(cid:12)(cid:12) X φ (cid:12)(cid:12)(cid:12) = D X r , X φ E = , i.e., X is a conformal immersion. Since X rr = , λ r cos φ, , − λ r sin φ ! , X φφ = (cid:18) − r cos φ, − λ r cos φ, − r sin φ, λ r sin φ (cid:19) , we get B ( X r , X r ) = λ r | X r | JX r , B (cid:16) X φ , X φ (cid:17) = − λ r | X r | JX r . In particular, Σ λ is a minimal Lagrangian surface in R .Notice that D X φ , JX E = . If 0 < | λ | < , then ∂ (cid:16) Σ λ ∩ B (cid:17) = Σ λ ∩ S has two components S ± ≔ ( r ± cos φ, λ r ± cos φ, r ± sin φ, − λ r ± sin φ ! : 0 ≤ φ < π ) , where r ± = s ± r − λ . These two components are Legendrian. The unit outward normal vector field of S ± ⊂ Σ λ ∩ B is ν ± = ± r ± cos φ, − λ r ± cos φ, r ± sin φ, λ r ± sin φ ! = √ − λ X ∓ λ JX . Thus, the contact angle θ ± along the boundary S ± satisfiessin θ ± = √ − λ , cos θ ± = ∓ λ. In summary, X is a conformal annulus minimal Lagrangian immersion from A = { ( r , φ ) : r − ≤ r ≤ r + , ≤ φ < π } to B with Legendrian capillary boundary on S with X ( A ) = Σ λ (0 < | λ | < ). Notice that the contact angle of Σ λ (0 < | λ | < ) can not be π .We have Theorem 3.3.
Assume that Σ is an annulus type minimal Lagrangian surface in B with Legendrian capillary bound-ary on S , then Σ must be congruent to one of the Lagrangian catenoids Σ λ (0 < | λ | < ) . roof. Assume that Σ is given by a conformal minimal immersion X from an annulus A = { ( r , φ ) : r − ≤ r ≤ r + , ≤ φ < π } ⊂ R to B , where we use polar coordinates ( r , φ ) on A . Denote by S ± ≔ { X ( r ± , φ ) : 0 ≤ φ < π } the boundary of Σ . Then z h B ( ∂ z , ∂ z ) , J ∂ z i = r h B ( ∂ r , ∂ r ) , J ∂ r i − √− r D B (cid:16) ∂ φ , ∂ φ (cid:17) , J ∂ φ E is holomorphic in Σ and the imaginary part vanishes on ∂ Σ and hence it vanishes on Σ . Therefore this holomorphicfunction must be a constant by maximum principle, which can not be zero (cf. Theorem 3.1). Consequently, there isa nonzero real constant c such that B ( ∂ r , ∂ r ) = c | ∂ r | r J ∂ r . When restricted on ∂ Σ = S + ∪ S − , according to (2.3), we have c = ∓ r | ∂ r | cos θ ± . By Lemma 2.2 we see that both S ± are Legendrian geodesics , and hence are Legendrian great circles on S . Conse-quently c = − cos θ + . Similarly, we have c = cos θ − . Thereforecos θ + + cos θ − = , sin θ + = sin θ − . Let λ ∈ ( − / , ∪ (0 , /
2) be the unique real number determined bysin θ ± = √ − λ , cos θ ± = ∓ λ. Since X is minimal we have ∆ g X = , where g = e u ( dr + r d φ ) is a conformal metric induced on X ( A ). Let ∆ be the metric on the flat annulus A , then ∆ X = . (3.2)Since both S ± are Legendrian great circles on S , there exist unit vectors ~ a ± , ~ b ± ∈ R with D ~ a ± , ~ b ± E = D ~ a ± , J ~ b ± E = S ± = ~ a ± cos φ + ~ b ± sin φ. (3.3)Then by applying the maximum principle to (3.2) with boundary conditions (3.3), we have X = X ( r , φ ) = ~ ar + λ~ br cos φ + ~ cr − λ~ dr sin φ, where ~ a , ~ b , ~ c , ~ d ∈ R are uniquely determined by θ ± , r ± and ~ a ± , ~ b ± . Direct computations show that X r = ~ a − λ~ br cos φ + ~ c + λ~ dr sin φ, φ = − ~ ar + λ~ br sin φ + ~ cr − λ~ dr cos φ. Thus | X r | − r (cid:12)(cid:12)(cid:12) X φ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ~ a − λ~ br (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) ~ c − λ~ dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cos φ + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ~ c + λ~ dr (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) ~ a + λ~ br (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin φ + * ~ a − λ~ br , ~ c + λ~ dr + + * ~ a + λ~ br , ~ c − λ~ dr + sin φ cos φ. It follows from | X r | − r (cid:12)(cid:12)(cid:12) X φ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ~ a (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ~ c (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) ~ b (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ~ d (cid:12)(cid:12)(cid:12)(cid:12) , D ~ a , ~ b E = D ~ c , ~ d E , (cid:10) ~ a , ~ c (cid:11) = , D ~ b , ~ d E = . (3.4)Then by (3.4) * X r , r X φ + = * ~ a − λ~ br , ~ c − λ~ dr + cos φ − * ~ c + λ~ dr , ~ a + λ~ br + sin φ + * ~ c + λ~ dr , ~ c − λ~ dr + − * ~ a − λ~ br , ~ a + λ~ br + sin φ cos φ = − λ r (cid:16)D ~ a , ~ d E + D ~ b , ~ c E(cid:17) , which implies from D X r , r X φ E = D ~ a , ~ d E + D ~ b , ~ c E = . (3.5)Moreover | X | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ~ ar + λ~ br (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cos φ + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ~ cr − λ~ dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin φ + * ~ ar + λ~ br , ~ cr − λ~ dr + sin φ cos φ. When restricted on the boundary S ± where r = r ± , we have | X | =
1, together with(3.4) and (3.5) we get (cid:12)(cid:12)(cid:12) ~ a (cid:12)(cid:12)(cid:12) r ± + λ (cid:12)(cid:12)(cid:12)(cid:12) ~ b (cid:12)(cid:12)(cid:12)(cid:12) r ± = , (3.6) D ~ a , ~ d E = D ~ b , ~ c E = D ~ a , ~ b E = D ~ c , ~ d E = . (3.7)In addition, since r h X r , X i = (cid:12)(cid:12)(cid:12) ~ a (cid:12)(cid:12)(cid:12) r − λ (cid:12)(cid:12)(cid:12)(cid:12) ~ b (cid:12)(cid:12)(cid:12)(cid:12) r , when restricted on the boundary S ± where r = r ± we have (cid:12)(cid:12)(cid:12) ~ a (cid:12)(cid:12)(cid:12) r ± − λ (cid:12)(cid:12)(cid:12)(cid:12) ~ b (cid:12)(cid:12)(cid:12)(cid:12) r ± = sin θ ± . (3.8)By (3.6) and (3.8), recall that sin θ ± = √ − λ , we obtain that (cid:12)(cid:12)(cid:12) ~ a (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ~ b (cid:12)(cid:12)(cid:12)(cid:12) = . η = (cid:12)(cid:12)(cid:12) ~ a (cid:12)(cid:12)(cid:12) >
0, by (3.4), (3.7) and (3.8) we have (cid:12)(cid:12)(cid:12) ~ a (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ~ c (cid:12)(cid:12)(cid:12) = η, (cid:12)(cid:12)(cid:12)(cid:12) ~ b (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ~ d (cid:12)(cid:12)(cid:12)(cid:12) = η , D ~ a , ~ d E = (cid:10) ~ a , ~ c (cid:11) = D ~ a , ~ b E = D ~ b , ~ c E = D ~ b , ~ d E = D ~ c , ~ d E = . (3.9)Moreover, * X r , r JX φ + = * ~ a − λ~ br , J ~ c − J λ~ dr + cos φ − * ~ c + λ~ dr , J ~ a + J λ~ br + sin φ + * ~ c + λ~ dr , J ~ c − J λ~ dr + − * ~ a − λ~ br , J ~ a + J λ~ br + sin φ cos φ. Therefore, by D X r , r JX φ E = (cid:10) ~ a , J ~ c (cid:11) = D ~ b , J ~ d E = , D ~ a , J ~ d E + D ~ b , J ~ c E = , D ~ a , J ~ b E + D ~ c , J ~ d E = . (3.10)Thus by (3.10) we have * r X , r JX φ + = * ~ a + λ~ br , J ~ c − J λ~ dr + cos φ − * ~ c − λ~ dr , J ~ a + J λ~ br + sin φ + * ~ c − λ~ dr , J ~ c − J λ~ dr + − * ~ a + λ~ br , J ~ a + J λ~ br + sin φ cos φ = λ r D ~ b , J ~ c E , which implies from D r X , r JX φ E = ∂ Σ and (3.10) that D ~ a , J ~ d E = D ~ b , J ~ c E = . In addition, r h X r , JX i = λ D ~ a , J ~ b E . When restricted on the boundary r = r ± , since r h X r , JX i = cos θ ± = ∓ λ, we conclude that D ~ a , J ~ b E = − . (3.11)Therefore, by (3.9), (3.10) and (3.11), the real metric O = (cid:16) ~ a ~ b ~ c ~ d (cid:17) satisfies O T O = η η η
00 0 0 η , O T JO = J = − − . Set Q = η η η
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