Few remarks on the Poincar\'e metric on a singular holomorphic foliation

Let $\mathcal{F}$ be a Riemann surface foliation on $M \setminus E$, where $M$ is a complex manifold and $E \subset M$ is a closed set. Assume that $\mathcal{F}$ is hyperbolic, i.e., all leaves of the foliation $\mathcal{F}$ are hyperbolic Riemann surface. Fix a hermitian metric $g$ on $M$. We will consider the Verjovsky's modulus of uniformization map $\eta$, which measures the largest possible derivative in the class of holomorphic maps from the unit disk into the leaves of $\mathcal{F}$. Various results are known to ensure the continuity of the map $\eta$ along the transverse directions, with suitable conditions on $M$, $\mathcal{F}$ and $E$. For a domain $U \subset M$, let $\mathcal{F}_{U}$ be the holomorphic foliation given by the restriction of $\mathcal{F}$ to the domain $U$, i.e., $\mathcal{F}\vert_{U}$. We will consider the modulus of uniformization map $\eta_{U}$ corresponding to the foliation $\mathcal{F}_{U}$, and study its variation when the corresponding domain $U$ varies in the Caratheodory kernel sense, motivated by the work of Lins Neto--Martins.