Two remarks on the Poincar\'{e} metric on a singular Riemann surface foliation

TL;DR

This paper investigates the behavior of the ratio between a fixed hermitian metric and the Poincaré metric on leaves of a foliation on a complex manifold, and extends a Bloch constant result to such foliations.

Contribution

It studies the variation of the ratio function of metrics on leaves and extends Minda's Bloch constant to Riemann surface foliations.

Findings

The ratio function ta_U varies continuously with the domain U.

A version of the Bloch constant is established for the foliation .

The results connect metric variation with hyperbolic geometry of leaves.

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 Poincar\'{e} 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 Poincar\'{e} 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 $\mathcal{F}$