Two remarks on the Poincaré metric on a singular Riemann surface foliation

Abstract

Let $\mathcal{F}$ be a smooth Riemann surface foliation on $M\setminus E$, where $M$ is a complex manifold and $E\subset M$ is a closed set. Fix a hermitian metric $g$ on $M\setminus E$ and assume that all leaves of $\mathcal{F}$ are hyperbolic. For each leaf $L\subset \mathcal{F}$, the ratio of $g|L$, the restriction of $g$ to $L$, and the Poincaré metric ${\lambda }_L$ on $L$ defines a positive function $\eta$ that is known to be continuous on $M\setminus E$ under suitable conditions on $M,E$. For a domain $U\subset M$, we consider ${\mathcal{F}}_U$, the restriction of $\mathcal{F}$ to $U$ and the corresponding positive function ${\eta }_U$ by considering the ratio of $g$ and the Poincaré metric on the leaves of ${\mathcal{F}}_U$. First, we study the variation of ${\eta }_U$ as $U$ varies in the Hausdorff sense motivated by the work of Lins Neto-Martins. Secondly, Minda had shown the existence of a domain Bloch constant for a hyperbolic Riemann surface $S$, which in other words shows that every holomorphic map from the unit disc into $S$, whose distortion at the origin is bounded below, must be locally injective in some hyperbolic ball of uniform radius. We show how to deduce a version of this Bloch constant for $\mathcal{F}$

11 pages