Lagrangians of Complex Minkowski Spaces Louis Yang Liu March 16, 2009 Abstract In this article, we mainly investigate the spaces of Cn with complex Finsler metrics and study their Lagrangian subspaces. For torus invariant complex Finsler metrics, we show that there is a topological torus of Lagrangians. For general complex Finsler metric, we show that the subspaces of Cn with complex Finsler metric in the orbit of any Lagrangian acted by the isometry group of the metric are Lagrangians.

A complex Finsler metrics is defined in many literatures, for example [A1], [AP] and [K], as follows Definition 1. A complex Finsler metric F is a function F : Cn → R that satisfies the following conditions 1. F (v) ≥ 0 for any v ∈ Cn and F (v) = 0 if and only if v = 0. 2. F (λv) = |λ|F (v) for any λ ∈ C and v ∈ Cn . 3. F is smooth on Cn \ {0}. Furthermore, a complex Finsler metric F is called to be peudoconvex if the (1.1)-real form 1√ ¯ 2) −1∂ ∂(F (1) 2 is positive-definite, see [A1], or equivalently the Hermitian matrix (gi¯j (ξ1 , · · · ξn ))n×n ,

(2)

in which gi¯j =

1 ∂ 2 (F 2 ) , 2 ∂ξi ∂ ξ¯j

(3)

is positive definite, see [M]. There is a class of complex Finsler metrics that attracts considerable interest, the torus invariant complex Finsler metric, which is defined as follows

1

Definition 2. A complex Finsler metric F on Cn is said to be torus invariant if F (eiθ1 ξ1 , · · · , eiθn ξn ) = F (ξ1 , · · · , ξn ) (4) for any (ξ1 , · · · ξn ) ∈ Cn and any (eiθ1 , · · · , eiθn ) ∈ Tn := U (1)n . A complex space Cn is called a complex Minkowski space if it has a complex Finsler metric F , denoted as (Cn , F ). In a complex Minkowski space (Cn , F ), the (1.1)-real form (1) gives a Kahler structure, and we’d like to find the Lagrangian subspaces under the Kahler structure. For this purpose, we are able to obtain the following theorem Theorem 3. The n-real planes in  Tn := span((eiθ1 , 0, · · · , 0), · · · , (0, · · · , 0, eiθn )) : (eiθ1 , · · · , eiθn ) ∈ U (1)n (5) are Lagrangians for any complex Minkowski space (Cn , F ) with torus invariant complex Finsler metric F . Proof. From (1), the Kahler form for the complex Minkowski space (Cn , F ) is κ=



−1

n X 1 ∂ 2 (F 2 ) dξi ∧ dξ¯j . 2 ∂ξi ∂ ξ¯j i,j=1

(6)

Let F (r1 , · · · , rn ) := F (|ξ1 |, · · · , |ξn |). The definition of torus invariant complex Finsler metric,(4), yields F (ξ1 , · · · , ξn )) = F (|ξ1 |, · · · , |ξn |).

(7)

Using the differential equalities

and

∂ 1 |ξi | |ξi | = ∂ξi 2 ξi

(8)

1 |ξi | ∂ |ξi | = ¯ 2 ξ¯i ∂ ξi

(9)

1 ∂ 2 (F 2 ) |ξi ξj | ∂ 2 (F 2 ) = . 4 ∂ri ∂rj ξi ξ¯j ∂ξi ∂ ξ¯j

(10)

for i = 1, · · · , n, we have

Now let’s take any ei0 := (0, · · · , eiθi0 , · · · , 0) and ej0 := (0, · · · , eiθj0 , · · · , 0), 1 ≤ i0 , j0 ≤ n, and evaluate the Kahler form κ on these two vectors at any point v = (λ1 eiθ1 , · · · , λn eiθn ) ∈ span((eiθ1 , 0, · · · , 0), · · · , (0, · · · , 0, eiθn )). It follows from (6) that √ ∂ 2 (F 2 ) i(θj −θi ) −1 ∂ 2 (F 2 ) i(θi −θj ) 0 0 − ( e e 0 0 ). (11) κv (ei0 , ej0 ) = 2 ∂ξi0 ∂ ξ¯j0 ∂ξj0 ∂ ξ¯i0 2

Applying (10), we obtain that √

κv (ei0 , ej0 )

=

λi0 λj0 −1 ∂ 2 (F 2 ) 8 ∂ri ∂rj ( λi eiθi0 λj e−iθj0 0 0 λj0 λi0



=

λj0 e

iθj 0

ei(θi0 −θj0 )

−iθi 0

λi0 e

ei(θj0 −θi0 ) )

(12)

0.

Therefore κ vanishes at every point on the n-plane span((eiθ1 , 0, · · · , 0), · · · , (0, · · · , 0, eiθn ))

(13)

in Tn , that finishes the proof. To generalize the results from complex Lp spaces, we would like to ask whether the n-real planes in  T1 := span((eiθ , 1, · · · , 1), · · · , (1, · · · , 1, eiθ )) : eiθ ∈ U (1) (14) are Lagrangians for any complex Minkowski space (Cn , F ) with torus invariant complex Finsler metric F or not. However, it turns out that the answer is no in general. For this, one can take the example of C2 with F (ξ1 , ξ2 ) = p 4 |ξ1 |4 + 2|ξ2 |4 . In this case, one can see that span(eiθ , 1), (eiθ , 1) is not a Lagrangian subspace under the Kahler form (6). For non-torus invariant complex Finsler metrics, we have the following remark to Theorem 3, Remark 4. In general , the n-real planes in  Tn := span((eiθ1 , 0, · · · , 0), · · · , (0, · · · , 0, eiθn )) : (eiθ1 , · · · , eiθn ) ∈ U (1)n (15) are not necessarily Lagrangian subspaces in complex Minkowski space (Cn , F ). One example is the following space, C2 with non-torus invariant complex Finsler metric q F (ξ1 , ξ2 ) = |ξ1 ξ¯2 + 2ξ¯1 ξ2 | + 2|ξ1 |2 + 2|ξ2 |2 . (16) In this example, a direct computation gives ∂ 2 (F 2 ) ∂ξ1 ∂ ξ¯2

=

1 (2|ξ1 ξ¯2 + 2ξ¯1 ξ2 |2 (5ξ2 ξ¯1 + 8ξ1 ξ¯2 ) 4|ξ1 ξ¯2 +2ξ¯1 ξ2 |3 2 −(5ξ¯1 |ξ2 |2 + 4ξ1 ξ¯2 )(5|ξ1 |2 ξ2 + 4ξ1 2¯ξ2 ))

(17)

and ∂ 2 (F 2 ) ∂ 2 (F 2 ) = . ∂ξ2 ∂ ξ¯1 ∂ξ1 ∂ ξ¯2

(18)

If we evaluate the Kahler form (6) on e1 := (0, eiθ1 ) and e2 := (0, eiθ2 ) at any point v = (λ1 eiθ1 , λ2 eiθ2 ) in the plane span(e1 , e2 ), we then have κv (e1 , e2 )

=

2 − |ei·2(θ1 −θ Im((2|ei·2(θ1 −θ2 ) + 2|2 + 5)ei(θ1 −θ2 ) 2 ) +2|3 +2ei·3(θ1 −θ2 ) )ei(θ1 −θ2 ) .

3

(19)

Let θ := θ1 − θ2 , then κv (e1 , e2 ) = −

2 Im(2|ei·2θ + 2|2 + 5)ei·2θ + 2ei·4θ . |ei·2θ + 2|3

(20)

One can see from (20)that the plane span(e1 , e2 ) must satisfy that θ1 − θ2 = kπ 2 , 2 k = 0, ±1, ±2, ±3 in order to have κ (e , e ) = 0. So in the torus T := v 1 2  span((eiθ1 , 0), (0, eiθ2 )) : (eiθ1 , eiθ2 ) ∈ U (1)2 we only have 4 circles of planes which are Lagrangian subspaces of the complex Minkowski space C2 with nontorus invariant complex Finsler metric (16). From the above discussions, we know that the Lagrangian subspaces depend greatly on Finsler metrics. Complex Minkowski spaces with Hermitian metrics as special Finsler metrics can have much more Lagrangian subspaces than complex Minkowski spaces with non-Hermitian Finsler metrics do, for instance complex L2 space and Lp space, 1 ≤ p < ∞, p 6= 2. Moreover, by perturbing some coefficients in Finsler metric one will have different Lagrangian subspaces for the space, for example, Lp(α,β) (ξ1 , ξ2 ) := (α|ξ1 |p + β|ξ2 |p )1/p , α, β ∈ R+ . Of course, for Lp(α,β) metric one can express the Lagrangians in terms of the parameters α and β, and the space of Lagrangians for Lp(α,β) will have the same topology as the one for Lp , but in the class of torus invariant complex Finsler metrics, one can have spaces of Lagrangians with very different topologies, though all the spaces of Lagrangians contain the torus (5). However, we can generalized Theorem 3 to general complex Finsler metrics in another direction. First we know that the space of Lagrangian subspaces in any complex Minkowski space is closed, and then Lemma 5. The space of Lagrangian subspaces in a complex Minkowski space (Cn , F ) is compact in Gr(n, Cn ). One can define an group G := {g ∈ GL(n, Cn ) : F (gv) = F (v) for any v ∈ Cn }

(21)

called the ismometry group of a Finsler metric F on Cn , see [A2]. Aikou in [A2] points out that one can use the method in [W] that treats real Finsler manifolds to show the following Lemma 6. The isometry group G of a complex Finsler metric is a compact Lie group. Based on the above lemma, we can establish the following theorem on the Lagrarians of a space with complex Finsler metric, for which thanks to Dr. Joseph H. G. Fu for pointing out a functorial proof, Theorem 7. Suppose an n-real plane P is a Lagrangian subspace of a complex Minkowski space (Cn , F ). Then for any g in the isometry group G the n-real plane gP which is P acted by g is also a Lagrangian subspace.

4

Proof. One can prove it by evaluating the Kahler form (6) on transformed vectors and carring out a direct computation. The other proof is functorial. Using the commutativity of the holomorphic ¯ in other words, g ∗ ∂ = ∂g ∗ map g with the complex differential operators ∂ and ∂, ∗¯ ∗ ¯ and g ∂ = ∂g , see [B], we have by (21) that g∗ κ = g∗ (

√ √ 1√ ¯ 2 )) = 1 −1∂ ∂(g ¯ ∗ F 2 ) = 1 −1∂ ∂F ¯ 2 = κ. −1∂ ∂(F 2 2 2

(22)

So if κ vanishes at any pair of vectors in P , it also does in gP .

References [A1] T. Aikou, Finsler geometry on complex vector bundles, MSRI Publications 50(2004), 85 - 107 [A2] T. Aikou, Complex manifolds modeled on a complex Minkowski space, J. Math. Kyoto Univ. 35 (1995), 85 - 103. [AP] M. Abate and G. Patrizio, Finsler metrics. A global approach, Springer Lecture Notes in Mathematics 1591, 1994. [B] Albert Boggess, CR manifolds and the tangential Cauchy-Riemann complex, Published by CRC Press, 1991 [K] Shoshichi Kobayashi, Negative vector bundles and complex Finsler structures, Nagoya Math. J. Volume 57 (1975), 153-166 [M] Gheorghe Munteanu, Complex spaces in Finsler, Lagrange, and Hamilton geometries, Published by Springer, 2004 [W] H. C. Wang, On Finsler spaces with completely integrable equations of Killing, J. London Math. Soc., (1947), 5-9

5

Lagrangians of Complex Minkowski Spaces

Mar 16, 2009 - Proof. From (1), the Kahler form for the complex Minkowski space (Cn,F) is κ = √. −1 n. ∑ i,j=1. 1. 2. ∂2(F2). ∂ξi∂ ¯ξj dξi ∧ d ¯ξj. (6). Let F(r1, ··· ,rn) := F(|ξ1|, ··· , |ξn|). The definition of torus invariant com- plex Finsler metric,(4), yields. F(ξ1, ··· ,ξn)) = F(|ξ1|, ··· , |ξn|). (7). Using the differential equalities. ∂. ∂ξi.

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