A uniformization theorem in complex Finsler geometry
aa r X i v : . [ m a t h . DG ] F e b A UNIFORMIZATION THEOREM IN COMPLEX FINSLERGEOMETRY
NINGWEI CUI ∗ , JINHUA GUO ∗∗ , LINFENG ZHOU † Abstract.
In complex Finsler geometry, an open problem is: does there exist aweakly K¨ahler Finsler metric which is not K¨ahler?In this paper, we give an affirmative answer to this open problem. More precisely,we construct a family of the weakly K¨ahler Finsler metrics which are non-K¨ahler. Theexamples belong to the unitary invariant complex Randers metrics. Furthermore,a uniformization theorem of the unitary invariant complex Randers metrics withconstant holomorphic curvature is proved under the weakly K¨ahler condition. Introduction
In Hermitian geometry, the vanishing of the torsion of the Hermitian connectionmeans that the metric is K¨ahler. In the complex Finsler geometry, since the torsion ofthe Chern-Finsler connection has a horizontal part and a mixed part, a complex Finslermetric to be K¨ahler is quite different. There are three kinds of notions which are thestrongly K¨ahler, K¨ahler and weakly K¨ahler metric corresponding to the vanishing ofthe different parts of the torsion of the Chern-Finsler connection [2].However, the relations of the three K¨ahler definition seem rather subtle. Chen andShen proved that a K¨ahler Finsler metric is actually a strong K¨ahler Finsler metric[3]. The well-known examples of smooth complex Finsler metrics: Kobayashi andCarath´eodory metrics, which agree on a bounded strictly convex domain, are weaklyK¨ahler metrics [2]. It is open whether they are K¨ahler or not. In [8], Xia and Zhongwrote: “ we also do not know whether there exists a weakly K¨ahler Finsler metric whichis not a K¨ahler Finsler metric. ” Thus the problem: does there exist a weakly K¨ahlerFinsler metric which is not K¨ahler , remains open.In this paper, we give an affirmative answer to this open problem. More precisely,we construct a family of the complex Randers metrics which are weakly K¨ahler butnot K¨ahler. Actually the following classification theorem is proved.
Theorem 1.1.
An unitary invariant complex Randers metric F = p rφ ( t, s ) definedon a domain D ⊂ C n , where r := | v | , t := | z | , s := |h z,v i| r , z ∈ D , and v ∈ T z D , is Mathematics Subject Classification.
Key words and phrases.
K¨ahler, weakly K¨ahler, holomorphic curvature, unitary invariant complexFinsler metric.The first author is supported by NSFC (No. 11401490), the third author is supported by theNational Key Research and Development Program of China under Grant 2018AAA0101001. weakly K¨ahler if and only if φ = (cid:16)r f ( t ) + tf ′ ( t ) − f ( t )2 t s + r tf ′ ( t ) + f ( t )2 t s (cid:17) or φ = f ( t ) + f ′ ( t ) s where f ( t ) is a positive smooth function. Remark 1.1. (1) When φ = f ( t ) + f ′ ( t ) s , it is easy to see that the metric F is aHermitian-K¨ahler metric.(2) In [9] , Zhong proved that if the unitary invariant complex Finsler metric F isK¨ahler if and only if it is Hermiatian-K¨ahler. When φ = (cid:16)r f ( t ) + tf ′ ( t ) − f ( t )2 t s + r tf ′ ( t ) + f ( t )2 t s (cid:17) , obviously, the complex Finsler metric F = p rφ ( t, s ) is not Hermitian, and thus F isweakly K¨ahler but not K¨ahler. As we know, the uniformization theorem in Hermitian geometry tells us that a com-plete, simple-connected K¨ahler manifolds of constant holomorphic (sectional) curvatureis isometric to one of the standard models: the Fubini-Study metric on C P n , the flatmetric on C n and the Bergmann metric on the unit ball in C n [5]. In complex Finslergeometry, one natural problem is to classify the weakly K¨ahler or K¨ahler Finsler metricwith constant holomorphic curvature.Roughly speaking, this problem is too ambitious to solve if one does not imposeany condition on the metrics. In [7], Xia and Zhong classified the unitary invariantweakly complex Berwald metrics of constant holomorphic curvature. However, if weassume the metrics are the unitary invariant Randers type, we can obtain a completeclassification theorem which is similar to the uniformization theorem in Hermitiangeometry. In this paper, we proved the following uniformization theorem. Theorem 1.2.
Let F = p rφ ( t, s ) be an unitary invariant complex Randers metricdefined on a domain D ⊂ C n , where r := | v | , t := | z | , s := |h z,v i| r . Assume F isnot a Hermitian metric. If F is a weakly K¨ahler Finsler metric and has a constantholomorphic curvature k if and only if (1) k = 4 , φ = ( q tc + t − t ( c + t ) s + q c ( c + t ) s ) defined on D = C n \ { } ; (2) k = 0 , φ = c ( √ t + √ s ) defined on D = C n ; (3) k = − , φ = q tc − t + t ( c − t ) s + q c ( c − t ) s defined on D = { z : | z | < √ c } where c is a positive constant. Remark 1.2.
It is hopeful to generalize above theorem to the more general complexRanders metrics.
The rest of this article is organized as follows. In section 2, we recall some notationsand formulas in complex Finsler geometry. In section 3, we introduce U and W tosimplify the weakly K¨ahler equation of the unitary invariant complex Finsler metrics. UNIFORMIZATION THEOREM IN COMPLEX FINSLER GEOMETRY 3
We also obtain the formula of the holomorphic curvature of the unitary invariantcomplex Finsler metrics under the weakly K¨ahler condition. In section 4, we give theexamples of the unitary invariant complex Randers metrics which are weakly K¨ahlerbut not K¨ahler and Theorem 1.2 is proved.2.
Preliminaries
Let M be an n -dimensional complex manifold. The canonical complex structure J acts on the complexified tangent bundle T C M so that T C M = T , M ⊕ T , M where T , M is called the holomorphic tangent bundle.The set { z , . . . , z n } is the local complex coordinate on M , where z α = x α + ix n + α ,1 ≤ α ≤ n , and { x , . . . , x n , x n +1 , . . . , x n } is the local real coordinate on M . Let ∂∂z α := 12 ( ∂∂x α − √− ∂∂x α + n ) , ∂∂ ¯ z α := 12 ( ∂∂x α + √− ∂∂x α + n ) . The set { ∂∂z , . . . , ∂∂z n } and { ∂∂ ¯ z , . . . , ∂∂ ¯ z n } are the local frames of T , M and T , M respectively. Thus for any ( z, v ) ∈ T , M , the vector v can be expressed as v = v α ∂∂z α .Similar to the Hermitian metric in complex geometry, the complex Finsler metric isdefined as the follows. Definition 2.1. (See [2] ) A complex Finsler metric F on a complex manifold M is acontinuous function F : T , M → [0 , + ∞ ) satisfying (1) G ( z, v ) = F ( z, v ) ≥ and G ( z, v ) = 0 if and only if v = 0 ; (2) G ( z, λv ) = | λ | G ( z, v ) for all ( z, v ) ∈ T , M and λ ∈ C ; (3) G is smooth on T , M , we say F is a smooth Finsler metric. Definition 2.2.
A complex Finsler metric F is called strongly pseudo-convex if theLevi matrix ( G α ¯ β ) = ( ∂ G∂v α ∂ ¯ v β ) is positive definite on T , M . The tensor G α ¯ β dz α ⊗ d ¯ z β is called the fundamentaltensor. There are several complex Finsler connections in complex Finsler geometry such asthe Chern-Finsler connection and the complex Berwald connection. For our conve-nience, we use the Chern-Finsler connection to derive the holomorphic curvature.For a function G ( z, v ) defined on T , M , we denote G α := ∂G∂v α , G ; α := ∂G∂z α , G α ; ¯ β := ∂ G∂v α ∂ ¯ z β . Let δδz α := ∂∂z α − N βα δδv β , δv α := dv α + N αβ dz β , N αβ := G α ¯ γ G ¯ γ ; β , then the tangent bundle of T , M splits into the horizontal and the vertical parts, i.e.: T C ˜ M = H ⊕ ¯ H ⊕ V ⊕ ¯ V NINGWEI CUI, JINHUA GUO, LINFENG ZHOU where ˜ M = T , M , H = span { δδz α } and V = span { ∂∂v α } . The Chern-Finsler connection1-forms are given by ω αβ = G α ¯ γ ∂G β ¯ γ = Γ αβ ; γ dz γ + C αβγ δv γ , where the connection coefficients can be written asΓ αβ ; γ := G α ¯ η δG β ¯ η δz γ , C αβγ := G α ¯ η ∂G β ¯ η ∂v γ . The curvature 2-forms Ω αβ areΩ αβ := ¯ ∂ω αβ = R αβ ; γ ¯ η dz γ ∧ d ¯ z η + S αβγ ;¯ η δv γ ∧ d ¯ z η + P αβ ¯ η ; γ dz γ ∧ δ ¯ v η + Q αβγ ¯ η δv γ ∧ δ ¯ v η , where R αβ ; γ ¯ η = − δ ¯ η (Γ αβ ; γ ) − C αβµ δ ¯ η ( N µγ ) ,S αβγ ;¯ η = − δ ¯ η ( C αβγ ) ,P αβ ¯ η ; γ = − ˙ ∂ ¯ η (Γ αβ ; γ ) − C αβµ ˙ ∂ ¯ η ( N µγ ) ,Q αβγ ¯ η = − ˙ ∂ ¯ η ( C αβγ ) . Definition 2.3. (See [2] ) The holomorphic curvature of a complex Finsler metric F is defined as K F ( v ) := 1 G G α R αβ ; γ ¯ η v β v γ ¯ v η = − G G α δ ¯ γ ( N αβ ) v β ¯ v γ . Similar to the Hermitian metric in complex geometry, the (2 , θ ( X, Y ) := ∇ X Y − ∇ Y X − [ X, Y ] , X, Y ∈ H ⊕ V . The K¨ahler condition means the above (2 ,
0) torsion is vanishing.
Definition 2.4. (See [2] ) Let F be a complex Finsler metric, and χ = v α δδz α ∈ Γ( H ) be the radial horizontal field. Then F is (1) strongly K¨ahler, if θ ( X, Y ) = 0 for any
X, Y ∈ H ; (2) K¨ahler, if θ ( X, χ ) = 0 for any X ∈ H ; (3) weakly K¨ahler, if h θ ( X, χ ) , χ i = 0 for any X ∈ H . In a local coordinate, the torsion θ is given by θ = (Γ αβ ; γ dz β ∧ dz γ + C αβγ δv β ∧ dz γ ) ⊗ δδz α . Hence F is a strongly K¨ahler metric if and only if Γ αβ ; γ = Γ αγ ; β ; F is a K¨ahler metricif and only if Γ αβ ; γ v γ = Γ αγ ; β v γ ; F is a weakly K¨ahler metric if and only if G α Γ αβ ; γ v γ = G α Γ αγ ; β v γ . UNIFORMIZATION THEOREM IN COMPLEX FINSLER GEOMETRY 5 Unitary invariant Finsler metrics
In complex Finsler geometry, there lack the canonical complex Finsler metrics com-paring to Hermitian geometry. In real Finsler geometry, the third author introducedthe spherically symmetric Finsler metrics which are invariant under any rotation in R n [10]. Similarly, in [9], Zhong introduced the unitary invariant complex Finsler metrics,which are invariant under any unitary action in C n , and gave an investigation on thecomplex Chern-Finsler connection and holomorphic curvature of the metrics. In thissection, we will recall some results in [9] and simplify some formulas by introducing U and W . Definition 3.1. (See [9] ) A complex Finsler metric F on a domain D ⊂ C n is calledunitary invariant if F satisfies F ( Az, Av ) = F ( z, v ) for any ( z, v ) ∈ T , M and A ∈ U ( n ) , where U ( n ) are the unitary matrices over thecomplex number field C . Theorem 3.1. (See [7] ) Let F be a strongly pseudo-convex complex Finsler metricdefined on a domain D ⊂ C n . The metric F is unitary invariant if and only if thereexists a smooth function φ ( t, s ) : [0 , + ∞ ) × [0 , + ∞ ) → (0 , + ∞ ) such that F ( z, v ) = p rφ ( t, s ) , r := | v | , t := | z | , s := |h z, v i| r for every ( z, v ) ∈ T , D . The fundamental tensor of F = p rφ ( t, s ) can be written as G α ¯ β = ( φ − sφ s ) δ α ¯ β + rφ ss s α s ¯ β + φ s ¯ z α z β . The determinant of the matrix ( G α, ¯ β ) isdet( G α, ¯ β ) = { ( φ − sφ s )[ φ + ( t − s ) φ s ] + s ( t − s ) φφ ss } ( φ − sφ s ) n − . Proposition 3.1. (See [9] ) The unitary invariant complex Finsler metric F = p rφ ( t, s ) is strongly pseudo-convex on a domain D ⊂ C n if and only if φ − sφ s > , ( φ − sφ s )[ φ + ( t − s ) φ s ] + s ( t − s ) φφ ss > . In [9], Zhong showed that a strongly pseudo-convex complex Finsler metric F = p rφ ( t, s ) defined on a domain D ⊂ C n is a K¨ahler Finsler metric if and only if φ ( t, s ) = a ( t ) + a ′ ( t ) s , where a ( t ) is a positive smooth function satisfying a ( t ) + ta ′ ( t ) >
0. Forthe weakly K¨ahler case, Zhong obtained the following result.
Theorem 3.2. (See [9] ) The unitary invariant complex Finsler metric F = p rφ ( t, s ) defined on a domain D ⊂ C n is weakly K¨ahler if and only if (3.1)( φ − sφ s )[ φ + ( t − s ) φ s ][ φ s − φ t + s ( φ st + φ ss )] + s ( t − s ) φ ss [ φ ( φ s − φ t ) + sφ s ( φ t + φ s )] = 0 . Actually, we can simplify the equation (3.1) and yield the following theorem.
NINGWEI CUI, JINHUA GUO, LINFENG ZHOU
Theorem 3.3.
An unitary invariant complex Finsler metric F = p rφ ( t, s ) definedon a domain D ⊂ C n is weakly K¨ahler if and only if (3.2) sU ( U − t ) W s − s ( U − t ) U s W − U − s ) U s = 0 where U = sφ + s ( t − s ) φ s φ and W = φ t + φ s φ .Proof. Let U := sφ + s ( t − s ) φ s φ , W := φ t + φ s φ . Then we have(3.3) φ s = U − ss ( t − s ) φ, φ t = ( W − U − ss ( t − s ) ) φ. It is easy to see φ − sφ s = t − Ut − s φ, φ + ( t − s ) φ s = Us φ.
Obviously, φ s − φ t = − W φ + 2 U − ss ( t − s ) φ and φ t + φ s = W φ , so that φ ( φ s − φ t ) + sφ s ( φ t + φ s ) = sW ( U − t ) + 2( U − s ) s ( t − s ) φ . By using φ st + φ ss = ( φ t + φ s ) s = W s φ + W φ s , we have φ s − φ t + s ( φ st + φ ss ) = sW s φ + U − tt − s W φ + 2( U − s ) s ( t − s ) φ. Moreover, from the first equation of (3.3), we compute φ ss = s ( t − s ) U s + U ( U − t ) s ( t − s ) φ. Substituting the above equalities into the weakly K¨ahler equation (3.1) implies theequation (3.2). (cid:3)
Lemma 3.1. U and W in Theorem 3.3 satisfy the following integral equation s ( U t + U s ) = s ( t − s ) W s + U. Proof.
From the formula of U and W , we can see that φ s = U − ss ( t − s ) φ, φ t = ( W − U − ss ( t − s ) ) φ. Then φ st = φ ts implies the result. (cid:3) Theorem 3.4.
Let F = p rφ ( t, s ) be the unitary invariant complex Finsler metricdefined on a domain D ⊂ C n and K F be the holomorphic curvature. Then in terms of φ , K F is K F ( v ) = − φ n [ s ( ∂k ∂t + ∂k ∂s ) + k ] + sφ + s ( t − s ) φ s φ [ s ( ∂k ∂t + ∂k ∂s ) + 2 k ] o , UNIFORMIZATION THEOREM IN COMPLEX FINSLER GEOMETRY 7 where k := ( φ − sφ s )[ φ + ( t − s ) φ s ] + s ( t − s ) φφ ss ,k := 1 k { [ φ + ( t − s ) φ s + s ( t − s ) φ ss ]( φ t + φ s ) − s [ φ + ( t − s ) φ s ]( φ st + φ ss ) } ,k := 1 k [ φ ( φ st + φ ss ) − φ s ( φ t + φ s )] . Proof.
According to the definition of the holomorphic curvature, we have K F = − G G α δ ¯ γ ( N αβ ) v β ¯ v γ = − G G γ δ ¯ ν (2 G γ )¯ v ν = − G G γ ∂∂ ¯ z ν (2 G γ )¯ v ν + 2 G G γ ¯ N αν ∂∂ ¯ v α (2 G γ )¯ v ν (3.4)where G γ := N γβ v β . For the unitary invariant complex Finsler metric F = p rφ ( t, s ),the complex spray coefficients are [9]2 G γ = N γβ v β = k h z, v i v γ + k h z, v i z γ . Therefore one needs to calculate ∂∂ ¯ z ν (2 G γ ) = ∂k ∂t h z, v i z ν v γ + ∂k ∂s sv ν v γ + k v ν v γ + ∂k ∂t h z, v i z ν z γ + ∂k ∂s s h z, v i v ν z γ + 2 k h z, v i v ν z γ and ∂∂ ¯ z ν (2 G γ )¯ v ν = r [ s ( ∂k ∂t + ∂k ∂s ) + k ] v γ + r [ s ( ∂k ∂t + ∂k ∂s ) + 2 k ] h z, v i z γ . Since G = F = rφ ( t, s ), we have G γ = ¯ v γ φ + rφ s s γ , where s γ := ∂s∂v γ = − r − ¯ v γ |h z, v i| + r − h z, v i ¯ z γ . Obviously, s γ v γ = 0 and s γ z γ = r − h z, v i ( t − s ). So we have − G G γ ∂∂ ¯ z ν (2 G γ )¯ v ν = − r φ (¯ v γ φ + rφ s s γ ) n r [ s ( ∂k ∂t + ∂k ∂s ) + k ] v γ + r [ s ( ∂k ∂t + ∂k ∂s ) + 2 k ] h z, v i z γ o = − φ n [ s ( ∂k ∂t + ∂k ∂s ) + k ]+ sφ + s ( t − s ) φ s φ [ s ( ∂k ∂t + ∂k ∂s ) + 2 k ] o . (3.5)On the other hand, it is necessary to compute ∂∂ ¯ v α (2 G γ ) = ∂∂ ¯ v α ( k h z, v i v γ + k h z, v i z γ )= ∂k ∂s s ¯ α h z, v i v γ + ∂k ∂s s ¯ α h z, v i z γ NINGWEI CUI, JINHUA GUO, LINFENG ZHOU and¯ N αν ∂∂ ¯ v α (2 G γ )¯ v ν = 2 ¯ G α ∂∂ ¯ v α (2 G γ )= 2( k h z, v i ¯ v α + k h z, v i ¯ z α )( ∂k ∂s s ¯ α h z, v i v γ + ∂k ∂s s ¯ α h z, v i z γ )= rs ( t − s ) k [ ∂k ∂s v γ + ∂k ∂s h z, v i z γ ] . Thus we obtain2 G G γ ¯ N αν ∂∂ ¯ v α (2 G γ )¯ v ν = 2 r φ (¯ v γ φ + rφ s s γ ) rs ( t − s ) k [ ∂k ∂s v γ + ∂k ∂s h z, v i z γ ]= 2 φ s ( t − s ) k [ ∂k ∂s + sφ + s ( t − s ) φ s φ ∂k ∂s ] . (3.6)Plugging (3.5) and (3.6) into (3.4) will yield K F ( v ) = − φ n [ s ( ∂k ∂t + ∂k ∂s ) + k ] + sφ + s ( t − s ) φ s φ [ s ( ∂k ∂t + ∂k ∂s ) + 2 k ] − s ( t − s ) k [ ∂k ∂s + sφ + s ( t − s ) φ s φ ∂k ∂s ] o . By a direct computation, we notice that ∂k ∂s + sφ + s ( t − s ) φ s φ ∂k ∂s = 0 . So the the formula of K F ( v ) holds. (cid:3) Remark 3.1. In [6] , Wang, Xia and Zhong also obtained the formula of the holomor-phic curvature of the unitary invariant complex Finsler metrics. When the metric is weakly K¨ahler, the holomorphic curvature has a nice formula interms of U and W . Theorem 3.5.
Let F be an unitary invariant metric defined on a domain D ⊂ C n .If F = p rφ ( t, s ) is a weakly K¨ahler Finsler metric, the holomorphic curvature K F isgiven by K F = − φ { s ( W t + W s ) − s ( t − s ) W s U s + W } where U = sφ + s ( t − s ) φ s φ and W = φ t + φ s φ .Proof. From U = sφ + s ( t − s ) φ s φ , W = φ t + φ s φ , one can obtain φ s = U − ss ( t − s ) φ, φ t = ( W − U − ss ( t − s ) ) φ. UNIFORMIZATION THEOREM IN COMPLEX FINSLER GEOMETRY 9
Substituting above equalities into the formulas of k , k and k in Theorem 3.4 derives k = U s φ , k = W U s − U W s U s , k = W s U s . When imposing the weakly K¨ahler condition i.e. sU ( U − t ) W s − s ( U − t ) U s W − U − s ) U s = 0 , one can simplify k and k to get k = − U − s ) s ( U − t ) , k = W − k U = WU + 2( U − s ) sU ( U − t ) . Thus ∂k ∂t = 2( t − s ) U t − U − s ) s ( U − t ) , ∂k ∂s = 2 s ( t − s ) U s + 2 U ( U − t ) s ( U − t ) ,∂k ∂t = ( WU ) t + U t U k − U ∂k ∂t , ∂k ∂s = ( WU ) s + U s U k − U ∂k ∂s . According to the formula of the holomorphic curvature K F in Theorem 3.4, theabove formulas involving k and k will imply K F = − φ n s ( W t + W s ) − sW ( U − t ) + 2( U − s ) U ( U − t ) ( U t + U s ) + 2 sW ( U − t ) + 2( U − s ) s ( U − t ) o . From the integral equation of U and W in Lemma 3.1, we have U t + U s = s ( t − s ) W s + Us .
Therefore the holomorphic curvature K F can be simplified as K F = − φ n s ( W t + W s ) − s ( t − s ) W W s U − s ( t − s ) 2( U − s ) U ( U − t ) W s + W o . Notice that the weakly K¨ahler condition tells us2( U − s ) U ( U − t ) = sW s U s − sWU . It leads to K F = − φ { s ( W t + W s ) − s ( t − s ) W s U s + W } , as we want in the Theorem. (cid:3) Weakly K¨ahler unitary invariant complex Finsler metrics withRanders type and an uniformization theorem
In 2009, Aldea and Munteanu initiated the study of the complex Randers motivatedby the notion of the real Randers metrics [1]. Later, Chen and Shen gave the formulaof the holomorphic curvature for complex Randers metrics and proved some rigiditytheorems on complex Randers metrics [4].
Definition 4.1.
A complex Finsler metric F on a complex manifold is called a complexRanders metric if F can be written as F = α + | β | , where α is a Hermitian metric and β is a (1 , -form, i.e. α = p a i ¯ j ( z ) v i ¯ v j , β = b i ( z ) v i . Remark 4.1. (1) The complex Randers metric is not smooth along the directions v = v i ∂∂z i with b i v i = 0 .(2) According to above definition, it is easy to see that an unitary invariant Finslermetric F = p rφ ( t, s ) is complex Randes metric if and only if φ = (cid:0)p f ( t ) + g ( t ) s + p h ( t ) s (cid:1) where f ( t ) , g ( t ) , h ( t ) are smooth functions and f ( t ) > , h ( t ) ≥ . Theorem 4.1.
An unitary invariant Randers metric F = p rφ ( t, s ) defined on adomain D ⊂ C n is a weakly K¨ahler Finsler metric if and only if φ = (cid:16)r f ( t ) + tf ′ ( t ) − f ( t )2 t s + r tf ′ ( t ) + f ( t )2 t s (cid:17) or φ = f ( t ) + f ′ ( t ) s where f ( t ) is a positive smooth function.Proof. The sufficiency is obvious by a direct computation.Now we prove the necessity. Since F = p rφ ( t, s ) is a weakly K¨ahler complexRanders metric, φ can be written as φ = (cid:0)p f ( t ) + g ( t ) s + p h ( t ) s (cid:1) where f ( t ) , g ( t ) , h ( t ) are smooth functions and f ( t ) > , h ( t ) ≥
0. Therefore U = sφ + s ( t − s ) φ s φ = s ( f + tg ) + t p s ( f + gs ) h √ f + gs ( √ f + gs + √ hs ) ,W = φ t + φ s φ = ( h + sh ′ ) √ f + gs + ( f ′ + g + sg ′ ) √ sh ( √ f + gs + √ hs ) p s ( f + gs ) h . Substituting above equalities into the equation of weakly K¨ahler condition (3.2), onewill get a quite complex equation as following { A ( t ) + A ( t ) p f + gs √ s + A ( t )( √ s ) + A ( t ) p f + gs ( √ s ) + A ( t )( √ s ) } f ( t ) = 0where A ( t ) = tf h ( tf ′ − f − tg ) . Notice that s = |h z,v i| | v | can be an arbitrary real number, thus A ( t ) = A ( t ) = A ( t ) = A ( t ) = A ( t ) = 0 . From A ( t ) = 0, we know that one of the following three equations holds: h ( t ) = 0 , f ( t ) = 0 or g ( t ) = tf ′ ( t ) − f ( t )2 t . UNIFORMIZATION THEOREM IN COMPLEX FINSLER GEOMETRY 11 If h ( t ) = 0, then the complex Randers metric F reduces to the Hermitian metricand then must be K¨ahler. It can imply φ = f ( t ) + f ′ ( t ) s. Since F is a complex Randers metric, f ( t ) = 0.If g ( t ) = tf ′ ( t ) − f ( t )2 t , substituting it into the equation of weakly K¨ahler condition againand rearranging all the terms, one can obtain { B ( t ) p f + gs √ s + B ( t )( √ s ) + B ( t ) p f + gs ( √ s ) + B ( t )( √ s ) } f ( t ) = 0where B ( t ) = t √ h ( tf ′ + f )( tf ′ + f − th ) . Therefore B ( t ) = B ( t ) = B ( t ) = B ( t ) = 0 . So B ( t ) = 0 deduces that one of the following three equations holds h ( t ) = 0 , tf ′ ( t ) + f ( t ) = 0 or h ( t ) = tf ′ ( t ) + f ( t )2 t . For the case of h ( t ) = 0, we have discussed previously. If tf ′ ( t ) + f ( t ) = 0 i.e. f ( t ) = ct with c a constant, then φ = (cid:0)r c ( t − s ) t + p h ( t ) s (cid:1) which means the Hermitian part q c ( t − s ) t is not positive defined. Therefore we have h ( t ) = tf ′ ( t ) + f ( t )2 t . (cid:3) Remark 4.2.
It is easy to check that if f ( t ) > and f ′ ( t ) > , the weakly K¨ahlercomplex Randers metric in above theorem is strongly pseudo-convex. Furthermore, we can prove the following uniformization theorem in complex Finslergeometry.
Theorem 4.2.
Let F be an unitary invariant complex Randers metric defined on adomain D ⊂ C n . Assume F is not a Hermitian metric. If F = p rφ ( t, s ) is a weaklyK¨ahler Finsler metric and has a constant holomorphic curvature k if and only if (1) k = 4 , φ = ( q tc + t − t ( c + t ) s + q c ( c + t ) s ) defined on D = C n \ { } ; (2) k = 0 , φ = c ( √ t + √ s ) defined on D = C n ; (3) k = − , φ = q tc − t + t ( c − t ) s + q c ( c − t ) s defined on D = { z : | z | < √ c } where c is a positive constant. Proof.
The sufficiency is obvious by a direct computation. We only need to prove thenecessity. By Theorem 4.1, the complex Randers metric F can be written as φ = (cid:16)r f ( t ) + tf ′ ( t ) − f ( t )2 t s + r tf ′ ( t ) + f ( t )2 t s (cid:17) . Then we have U = s ( f + tg ) + t p s ( f + gs ) h √ f + gs ( √ f + gs + √ hs ) , W = ( h + sh ′ ) √ f + gs + ( f ′ + g + sg ′ ) √ sh ( √ f + gs + √ hs ) p s ( f + gs ) h where g = tf ′ ( t ) − f ( t )2 t and h = tf ′ ( t )+ f ( t )2 t .If the holomorphic curvature k = 0, one can get K F = − φ { s ( W t + W s ) − s ( t − s ) W s U s + W } = 0 . Plugging U and W into the above equation will yield A ( t ) p f + gs + A ( t ) √ s + · · · + A ( t )( √ s ) = 0where A ( t ) = t √ hf ( tf ′ + f )( tf ′ − f ). Hence A ( t ) = 0. Since F is not a Hermitianmetric, we know that tf ′ − f = 0 . It implies that f ( t ) = ct and φ = c ( √ t + √ s ) .If the holomorphic curvature k = 4, one can get K F = − φ { s ( W t + W s ) − s ( t − s ) W s U s + W } = 4 . Plugging φ , U and W into the above equation will yield B ( t ) p f + gs + B ( t ) √ s + · · · + B ( t )( √ s ) = 0where B ( t ) = t √ hf ( tf ′ + f )( tf ′ + 2 tf − f ). Hence B ( t ) = 0. From it, we knowthat tf ′ + 2 tf − f = 0 . It implies that f ( t ) = tc + t and φ = ( q tc + t − t ( c + t ) s + q c ( c + t ) s ) .If the holomorphic curvature k = −
4, one can get K F = − φ { s ( W t + W s ) − s ( t − s ) W s U s + W } = − . Plugging φ , U and W into the above equation will yield C ( t ) p f + gs + C ( t ) √ s + · · · + C ( t )( √ s ) = 0where C ( t ) = t √ hf ( tf ′ + f )( tf ′ − tf − f ). Hence C ( t ) = 0. From it, we knowthat tf ′ − tf − f = 0 . It implies that f ( t ) = tc − t and φ = q tc − t + t ( c − t ) s + q c ( c − t ) s . (cid:3) UNIFORMIZATION THEOREM IN COMPLEX FINSLER GEOMETRY 13
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80 (2012), no. 1-2,67-77. ∗ Department of Mathematics, Southwest Jiaotong University, Chengdu, 610031,P.R. China
Email address : [email protected] ∗∗ School of Mathematics Science, East China Normal University, Shanghai, 200241,P.R. China
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