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 η 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 $\mathcal{F}$.

让$\mathcal{F}$是一个光滑的黎曼表面叶状体$米\setminus 乙$，其中M是复流形，并且$乙\subset 米$是闭集。修正厄米度量g$米\setminus 乙$并假设所有的叶子$\mathcal{F}$是双曲线的。对于每一片叶子$\mathrm{大号}\subset \mathcal{F}$, 的比率$G|\mathrm{大号}$, g对L的限制, 和 Poincaré 度量${\lambda }_{\mathrm{大号}}$在L上定义了一个正函数η，它已知是连续的$米\setminus 乙$在M , E的合适条件下。对于域$ü\subset 米$， 我们认为${\mathcal{F}}_{ü}$, 的限制$\mathcal{F}$到U和相应的正函数${\eta }_{ü}$通过考虑g的比率和叶子上的 Poincaré 度量${\mathcal{F}}_{ü}$. 首先，我们研究变化${\eta }_{ü}$因为U在 Hausdorff 意义上的变化是由 Lins Neto-Martins 的工作所激发的。其次，Minda 已经证明了双曲黎曼曲面S的域布洛赫常数的存在，换句话说，这表明从单位圆盘到S的每个全纯映射，其在原点的失真在下面有界，在某些情况下必须是局部内射的均匀半径的双曲线球。我们展示了如何推导出这个 Bloch 常数的版本$\mathcal{F}$.