Schwarz Lemma and Hartogs Phenomenon in Complex Finsler Manifold

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1 Chin. Ann. Math. 34B3), 2013, DOI: /s Chinese Annals of Mathematis, Series B The Editorial Offie of CAM and Springer-Verlag Berlin Heidelberg 2013 Shwarz Lemma and Hartogs Phenomenon in Complex Finsler Manifold Bin SHEN 1 Yibing SHEN 1 Abstrat The authors prove the Shwarz lemma from a ompat omplex Finsler manifold to another omplex Finsler manifold and any omplete omplex Finsler manifold with a non-positive holomorphi urvature obeying the Hartogs phenomenon. Keywords Complex Finsler manifold, Shwarz lemma, Hartogs phenomenon 2000 MR Subjet Classifiation 51M11, 53C24 1 Introdution The Shwarz lemma is a result of omplex analysis of holomorphi funtions from the open unit disk to itself. Although the lemma is less elebrated than stronger theorems, suh as the Riemann mapping theorem, it is one of the simplest results apturing the rigidity of holomorphi funtions. Pik made a new expansion of Shwarz lemma, and Ahlfors first introdued the geometry onept, urvature into the omplex analysis see [1]). In 1978, Yau generalized it into the Kähler manifold see [2]). Sine it is useful in both analysis and geometry, we onsider the holomorphi map of the Finsler manifold. In the general ase, we need the manifold to be ompat beause the general maximum priniple is not admitted. The Hartogs phenomenon was first presented as the Hartogs extension theorem, whih is a fundamental result in the theory of funtions of several omplex variables. Informally, it states that the support of the singularities of suh funtions annot be ompat. This property of holomorphi funtions is also alled Hartogs phenomenon see [3]). Griffiths and Shiffman generalized it to the Hermitian manifold see [4 5]). We also onsider it in the Finsler ase. Given a Finsler urvature ondition, the phenomenon is kept. The method whih we use in those problems is based on the definition of the holomorphi urvature in Finsler geometry see [6]). By this definition, we an also obtain the embedding theorem as a orollary. 2 The Preliminaries Given a omplex manifold M, the real tangent bundle of M is denoted by T R M. T 1,0 M denotes the holomorphi tangent bundle of M and M is T 1,0 M\{0}. If the omplex dimension of M is n, let{z 1,,z n } be a set of loal omplex oordinates, where z α = x α +ix n+α. x 1,,x n,x n+1,,x 2n ome to be loal real oordinates. A loal frame over R for T R M is given by { o 1,, 2n, o o 1,, 2n}, o where o a = x and o a a = u. Analogously, a loal frame a Manusript reeived November 21, Revised July 18, Department of Mathematis, Southeast University, Nanjing , China; Department of Mathematis, Zhejiang University, Hangzhou , China. shenbin gs@hotmail.om yibingshen@zju.edu.n Projet supported by the National Natural Siene Foundation of China No ) and the Dotoral Program Foundation of the Ministry of Eduation of China No ).

2 456 B. Shen and Y. B. Shen over C for T 1,0 M is given by { 1,, n, 1,, n }, where α = z and α α = v. In the α above, we denote by z α = x α +ix n+α the oordinates on the manifold and v α = u α +iu n+α the vetors on M. Moreover, we use Roman indies to run from 1 to 2n, whereas Greek indies to run from 1 to n. Thenany1, 0)-vetor an be written as v = v α z. {z α ; v α } is a group of α loal omplex oordinates on T 1,0 M. We define as follows. Definition 2.1 A omplex Finsler metri on a omplex manifold M is a ontinuous funtion F : T 1,0 M R + satisfying a) G = F 2 is smooth on M. b) F v) > 0 for any v M. ) F λv) = λ F v) for any v T 1,0 M and λ C. We all a omplex manifold endowed with suh a metri a omplex Finsler manifold. Moreover, if the Levi matrix G αβ := α β G) is positively defined, we all F is strongly pseudoonvex. Generally, the Caratheodory and Kobayashi metris are strongly pseudoonvex in the strongly onvex domain see [7]). G is smooth on the whole of T 1,0 M, if and only if F is the norm assoiated to a Hermitian metri. In this ase, we shall say that F omes from a Hermitian metri see [6]). From the projetive map π : TM M, the definition of the real vertial bundle is given by while the omplexified vertial bundle is V R =ker π T R M, V C = V R R C =ker π T C M. Here the π ommutes with the omplex struture J, for the projetion π is holomorphi. V C splits into V C = V 1,0 V 0,1, and we define the omplex vertial bundle as V = V 1,0 =ker π T 1,0 M. The omplex horizontal bundle is a omplex subbundle H C T C M, whihisj-invariant, a onjugation invariant, suh that T C M = HC V C. Sine H C is also J-invariant, we an write H C = H 1,0 H 0,1, where H 1,0 = H C T 1,0 M. Moreover, H 1,0 = H 0,1, whih means that a omplex horizontal bundle is ompletely determined by its 1, 0)-part H 1,0.WesimplydenoteH 1,0 as H. Loally, we shall denote by indies like α, β and so on the derivatives with respet to the v-oordinates, for example, G αβ = 2 G. On the other hand, the derivatives with respet to v α v β the z-oordinates will be denoted by indies after a semiolon, for instane, G ;μν = G α;ν = 2 G z ν v. Levi matrix indues the fundamental tensor G α αβ dz α dz β. Setting 2 G z μ z ν δ δz α := z α N α β v β, δvα := dv α + Nβ α dz β, Nβ α := G αγ G γ;β, we have T C M = H H V V, where H =span { } { δ δz, V =span } α v. α To any Hermitian metri, there is an assoiated unique omplex linear onnetion suh that the metri tensor is parallel: the Chern onnetion. Analogously, to any strongly pseudoonvex Finsler metri, there is a unique good omplex vertial onnetion D making the Hermitian or

3 Shwarz Lemma and Hartogs Phenomenon in Complex Finsler Manifold 457 struture parallel. This onnetion is the Chern-Finsler onnetion. Preisely, G αβ dz α dz β is a Hermitian metri on the bundle H, i.e., δ α,δ β := G αβ. The Hermitian onnetion of this metri is alled the Chern-Finsler onnetion of a omplex Finsler metri. The onnetion 1-form is see [6]) ωβ α := Gαγ G βγ =Γ α β;η dzη + Cβη α δvη, where Γ α β;η := G αγ δ η G βγ, Cβη α := G αγ G βγη. 3 The Holomorphi Curvature Let us onsider the Finsler manifold M,F) now. The urvature form of the Chern-Finsler onnetion an be expressed as Ω α β := R α β;γηdz γ dz η + S α βγ;ηδv γ dz η + P α βη;γdz γ δv η + Q α βγηδv γ δv η. 3.1) Moreover, we set R αβ;γη := G μβ R μ α;γη, where ; denote the vertial derivative and G = F 2. The holomophi urvature on a Finsler manifold M,F) is defined as follows. Definition 3.1 The holomophi urvature K F v) along the diretion v is K F v) =K F χv)) = Ωχ, χ)χ, χ v Gv) 2. Here v T 1,0 M\{0} and χ is the horizontal lifting from T 1,0 M to HTM. Under the loal oordinates, we rewrite it as K F v) = 1 G 2 G αδ β Γ α ;γ )vγ v β = 1 G 2 δ β G ;γ)v γ v β. When F is Hermitian metri, K F is the holomorphi urvature of the Hermitian metri. So the first term in the urvature form is a Hermitian quantum, while the other three terms are the non-hermitian quanta. Abate and Patrizio gave the following equation: Kϕ G)0) = K F v) 2 ϕ Gv) 2 ) H ϕ ) H ϕ ) H,χ ϕ )H v χ 2 χ, χ v v, where ϕ :Δ M is a holomorphi map from the unit disk in C to the manifold M. Based on it, they gave another analyti expression see [6]) K F v) =supk F [v]). 3.2) The supremum is taken in all the Gauss surfaes that are tangent to v at p. Remark 3.1 This definition is first given in [6, p. 110, p. 144]. The supremum in their definition is taken in the family of holomorphi maps φ from the unit disk in C to the manifold M with φ0) = p and φ 0) = λv for λ C. Suh families are just the omplex 1-dimensional submanifolds passing point p with diretion v. The omplex 1-dimensional submanifolds of a Finsler manifold are Gauss surfaes. So the definition here is the same as the one given in [6]. When it is on a Riemannian surfae, suh definition is just the Gauss urvature. Deriving from the definition, we know that the following proposition holds. Proposition 3.1 The holomorphi urvature of the submanifold of any Finsler manifold is delined.

4 458 B. Shen and Y. B. Shen By the urvature delined ondition, we see that there must be a setion of negative urvature on a manifold of negative holomorphi urvature. We have the following result. Theorem 3.1 A ompat omplex Finsler manifold with negative holomorphi urvature must be a Hodge manifold. Proof By Proposition 3.1, there is a line bundle with negative urvature on a negative urved Finsler manifold. The dual bundle is positive. The lassial Kodaira embedding theorem implies that a ompat omplex manifold admitting a positive line bundle an be embedded into CP n holomorphially, whih is independent of the metri. By this, suh a manifold must be a Hodge manifold. 4 The Shwarz Lemma Using 3.2), we an prove the following Shwarz lemma. Theorem 4.1 Let M 1,F 1 ), M 2,F 2 ) be two Finsler manifolds, and M 1,F 1 ) be ompat. Their orresponding holomorphi urvatures satisfy K F1 B,K F2 A for A, B > 0. Let ϕ be the holomorphi map from M 1 to M 2.Thenϕ F2 2 B A F 1 2. Proof For i =1, 2, K Fi =supk Fi [v]). Let ϕ F2 2 = μp, v)f 1 2 for a positive funtion μ C depending on p and v. Sine the restrition of metri F 1 on the Gauss surfae is a Hermitian metri, we set F1 2 = λdzdz, wherez is the omplex oordinate on. As the setting, )) ϕ F2 2 = μf1 2 = μ ξz),ξ λdzdz, z where ξ is a holomorphi map from a Gauss surfae to M, suh that p = ξz) andv = ξ z ). By the definition of Gauss urvature on the Gauss surfae, we see sup K ϕ F 2 [v]) = sup Δ z log μ +Δ z log λ ). μλ The Laplaian is taken on. Sineμ takes the maximum on PTM, whih is ompat sine M 1 is ompat, we have μξz),ξ z )) μξ0),ξ z 0)) for ξ from the unit disk in C to M with ξ0) = p 0, ξ z 0)=v 0 and max μ = μp 0,v 0 ). It gives Δ z log μ0) 0. PTM By the definition, the urvatures an be written as K F1 v) =sup{k F1 [v])} =sup{kψ F1 2 )0)}, where ψ is the holomorphi map from to M 1 with ψ0) = p, ψ z = v). K ϕ F 2 v) =sup{k ϕ F 2 [v])} =sup{kψ ϕ F2 2 )0)}, K F2 ϕ v)=sup{k F2 [ϕ v])} =sup{kξ F2 2 )0)}, where ξ is the holomorphi map from to M 2 with ξ0) = ϕp), ξ z = ϕ v ). It is easy to see K ϕ F 2 v) K F2 ϕ v), and it follows immediately that 1 μ 0 sup Δ z log λ λ ) A, where μ 0 is the maximum of μ and is independent of. By the definition of K F1 and the assumption, it gives ϕ F2 2 μ 0F1 2 B A F 1 2. It immediately gives the following orollary.

5 Shwarz Lemma and Hartogs Phenomenon in Complex Finsler Manifold 459 Corollary 4.1 Let M 1,g) be a ompat Hermitian or Sasakian) manifold, and M 2,F 2 ) be a Finsler manifold. Their orresponding holomorphi urvatures or transverse holomorphi urvatures) satisfy K 1 B,K F2 A for A, B > 0. Let ϕ be the holomorphi or Φ,J)- holomorphi) map from M 1 to M 2.Thenϕ F2 2 B A g or ϕ F2 2 B A gt ). Proof The Hermitian ase is obvious, and now we see the Sasakian one. Sine ϕ is Φ,J)- holomorphi, ϕ F 2 )X) =F 2 dϕx)). Taking X to be ξ, wegetϕ F 2 )ξ) =0,whihmeans that the pull bak metri ϕ F 2 just depends on the transverse metri g T. Remark 4.1 The Sasakian manifold is an odd dimensional one, whih is onsidered as the twin sister of the Kähler manifold. One an refer to [8 9] for more details. 5 The Hartogs Phenomenon The Hartogs phenomenon is from the Hartogs extension theorem. The theorem shows that, when n 2and0 a < b, any holomorphi funtion defined in a spherial shell Da,b n = {z Cn a< z 2 <b} an be extended to the ball Bb n of radius entered at the origin). In other words, there exists a holomorphi funtion on Bb n whose restrition on Dn a,b is just f. In general, we have the following definition see [10]). Definition 5.1 A omplex manifold M n is said to obey the Hartogs phenomenon, if for any 1 >a 0, any holomorphi map from Da,1 2 into M an be extended to a holomorphi map from the unit ball B 2 into M. Aording to [10], if M obeys the Hartogs theorem, then for m 2andany0 a<b, any holomorphi map from Da,b m into M an be extended to a holomorphi map from Bm b into M. One may restrit suh a map to the intersetion of a 2-dimensional omplex plane with the spherial shell, and then apply the definition. There are omplex manifolds that do not obey the Hartogs phenomenon. For example, when n 2, C n \{0} does not obey it, and any Hopf manifold does not obey it, either. Griffiths and Shiffman proved the following theorem see [10]). Theorem 5.1 Griffiths-Shiffman) Any omplete Hermitian manifold with non-positive holomorphi setional urvature obeys the Hartogs phenomenon. Based on the proof of the above theorem by Griffiths and Shiffman, we an prove the following theorem. Theorem 5.2 Any omplete omplex Finsler manifold with non-positive holomorphi urvature obeys the Hartogs phenomenon. Proof Let M n,f) be suh a manifold. Let f be the holomorphi map from D := Da,1 2 to M, whereda,1 2 = {z C2 a< z 2 < 1}. By the non-positive holomorphi urvature assumption, with the same argument, it follows that { 0 K f F v) =sup Δ z ln μ +Δ z ln λ }. μλ Notiing μ>0, for any diretion fixed, Δ z ln μ λ sup It means that for any fixed diretion v, wehave Δμ 0. Δ z ln λ ) =0. λ

6 460 B. Shen and Y. B. Shen When we fix v, μz,v) is a subharmoni funtion on D. Nextwewanttoshowthatforany ε>0, there is a onstant C>0, suh that f F 2 Cg 0,whereg 0 is the Hermitian metri on D. Fix a 1, suh that a<a 1 < 1 ε, and denote D = Da,1 ε, 2 D = Da 2 1,1 ε. Let C be sup μ< sine D is ompat, and so does SD. SD For any p in D with p a 1, it follows that μ C. When p a 1,letP be the omplex line path through point p and orthogonal to the line onneting 0 and p. We see that P D is a dis of p, while P D is an annulus. Take a loop γ in D around p. We have that for any fixed v, μp, v) 1 γ μdv C. With these two ases, we have that for a fixed diretion γ v, f F 2 Cg 0. Any sequene approahing p S a in the radial diretion is a Cauhy sequene. For any p k, p l in D an be jointed by a line with the tangent vetor v. Then it follows that dfp k ),fp l )) Cd 0 p k,p l ). fp k ) onverges to a point in M by the ompliity of M. Then, any holomorphi map f from D to M an be extended on the inner boundary S a of D. Fixing p S a,there is a neighborhood U D, suh that U is an open neighborhood of p in D. f is given by n holomorphi funtions on U. The lassial Hartogs theorem shows that f anbeextendedonto Da 2,1 with a <a. Aknowledgements The authors would like to express gratitude to Prof. F. Y. Zheng for his areful hek, and thank the referees for their suggestions. Referenes [1] Ahlfors, L. V., Complex Analysis, MGraw-Hill, Aukland, [2] Yau, S. T., A general Shwarz lemma for Kahler manifolds, Amer. J. Math., 1001), 1978, [3] Griffiths, P. and Harris, J., Priniples of Algebrai Geometry, Wiley-Intersiene, New York, [4] Griffiths, P., Two theorems on extension of holomorphi mappings, Invent. Math., 14, 1971, [5] Shiffman, B., Extension of holomorphi maps into Hermitian manifolds, Math. Ann., 194, 1971, [6] Abate, M. and Patrizio, G., Finsler Metri A Global Approah, Leture Notes in Math., 1591, Springer- Verlag, Berlin, Heidelberg, [7] Lempert, L., A metrique de Kobayashi et la representation des domains sur la boule, Bull.So.Math. Frane, 109, 1981, [8] Boyer, C. P. and Galiki, K., Sasakian geometry, holonomy and supersymmetry, to appear. arxiv: math/ [9] Boyer, C. P. and Galiki, K., On Sasakian-Einstein geometry, Intenat. J. Math., 11, 2000, [10] Zheng, F., Complex Differential Geometry, Study in Advaned Soiety, Vol. 18, A. M. S., Providene, RI, 2000.