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

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 $\lambda _{g}$, with respect to some metric $g$, continuous? It is known that the metric $\frac{g}{4\lambda _{g}}$ induces the Poincaré metric on the leaves. 2. When is the metric $\frac{g}{4\lambda _{g}}$ 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 $\lambda _{g}$ is continuous and $\frac{g}{4\lambda _{g}}$ 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 $g$, $\frac{g}{4\,\lambda _{g}}$ is not complete (§3.2).