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In this paper we consider the problem of uniformization of the leaves of a holomorphic foliation by curves in a complex manifold M. We consider the following problems: 1. When is the uniformization function λ g , with respect to some metric g, continuous? It is known that the metric g/4λ g induces the Poincare metric on the leaves. 2. When is the metric g/4λ g complete? We extend the concept of ultra-hyperbolic metric, introduced by Ahlfors in 1938, for singular foliations by curves, and we prove that if there exists a complete ultra-hyperbolic metric g. then λ g is continuous and g/4λ g is complete. In some local cases we construct such metrics, including the saddle-node (Theorem 1) and singularities given by vector fields with the first non-zero jet isolated (Theorem 2). We also give an example where for any metric g, g/4λ g is not complete ( 3.2).

Hermitian metrics inducing the Poincaré metric, in the leaves of a singular holomorphic foliation by curves