  We introduce a method to estimate the size of the domain of definition of the
solutions of a meromorphic vector field on a neighborhood of its pole divisor.
The corresponding techniques are, in a certain sense, quantitative versions of
some well-known phenomena related to the presence of metrics with positive
curvature. Several applications of these ideas are provided including a type of
"confinement theorem" for solutions of complete polynomial vector fields on
$\mathbb{C}^n$ and obstructions for certain (germs of) vector fields to be
realized by a global holomorphic vector field on a compact K\"ahler manifold.
As a complement a new approach to certain classical equations is proposed and
detailed in the case of Halphen equations.
