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.
holomorphic foliations Poincaré metric on the leaves uniformization of the leaves