Uniformization and the Poincaré metric on the leaves of a foliation by curves

  • A. Lins Neto
Article

DOI: 10.1007/BF01241634

Cite this article as:
Neto, A.L. Bol. Soc. Bras. Mat (2000) 31: 351. doi:10.1007/BF01241634

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 (αn)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.

Keywords

holomorphic foliations Poincaré metric on the leaves uniformization of the leaves 

Copyright information

© Sociedade Brasileira de Matemática 2000

Authors and Affiliations

  • A. Lins Neto
    • 1
  1. 1.Instituto de Matemática Pura e Aplicada Estrada Dona CastorinaRio de JaneiroBrasil