Abstract
In this paper we prove that a holomorphic foliation by curves, on a complex compact manifoldM, whose singularities are non degenerated and whose tangent line bundle admits a metric of negative curvature, satisfies the following properties:(a): All leaves are hyperbolic.(b): The Poincaré metric on the leaves is continuous.(c): The set of uniformizations of the leaves by the Poincaré disc D is normal. Moreover, if (α n ) n ≥1 is a sequence of uniformizations which converges to a map α: D, then either α is a constant map (a singularity), or α is an uniformization of some leaf. This result generalizes Theorem B of [LN], in which we prove the same facts for foliations of degree ≥2 on projective spaces.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
[Ah-1] L. V. Ahlfors,Conformal Invariants. Topics in Geometric Function Theory. McGraw-Hill (1973).
[Ah-2] L. V. Ahlfors,An Extension of Schwarz' Lemma. Trans. Am. Math. Soc.,43 (1938), 359–364.
[Br] M. Brunella,Feuilletages Holomorphes sur les Surfaces Complexes Compactes. Ann. Scient. Éc. Norm. Sup., 4e série, t.30 (1997), 569–594.
[C] A. Candel,Uniformization of Surface Laminations. Ann. Sc. École Norm. Sup.,26(4) (1993), 489–515.
[F] R. Friedman,Algebraic Surfaces and Holomorphic Vector Bundles. Universitext, Springer, (1998).
[G] A. A. Glutsyuk,Hyperbolicity of Leaves of a Generic One-Dimensional Holomorphic Foliation on a Nonsingular Projective Algebraic Manifold. Proc. of the Steklov Institute of Mathematics,213 (1996), 83–103.
[G-H] Griffiths-Harris,Principles of Algebraic Geometry. John-Wiley and Sons, (1994).
[LN] A Lins Neto,Simultaneous Uniformization for the Leaves of Projective Foliations by Curves. Bol. da Sociedade Brasileira de Matemática,25(2) (1994), 181–206.
[Se] A. Seidenberg,Reduction of singularities of the differential equation Ady=Bdx. Amer. J. de Math.,90 (1968), 248–269.
[V] A. Verjovsky,An Uniformization Theorem for Holomorphic Foliations. Contemp. Math.,58(III) (1987), 233–253.
Author information
Authors and Affiliations
Additional information
This research was partially supported by Pronex-Dynamical Systems, FINEP-CNPq.
About this article
Cite this article
Neto, A.L. Uniformization and the Poincaré metric on the leaves of a foliation by curves. Bol. Soc. Bras. Mat 31, 351–366 (2000). https://doi.org/10.1007/BF01241634
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01241634