论文标题
关于单数riemann表面叶面上的庞加莱度量标准的两句话
Two remarks on the Poincaré metric on a singular Riemann surface foliation
论文作者
论文摘要
令$ \ Mathcal {f} $为$ M \ setMinus e $上的平滑Riemann Surface Foliation，其中$ m $是一个复杂的歧管，$ e \ subset m $是封闭的集合。修复$ m \ setminus e $上的Hermitian公制$ g $，并假设所有$ \ Mathcal {f} $的叶子都是夸张的。对于每个叶子$ l \ subset \ mathcal {f} $，$ g |的比率l $，$ g $ to $ l $的限制，$ l $上的庞加莱公制$λ_l$定义了一个正函数$η$，在$ m，e $的适当条件下，众所周知，在$ m \ setminus e $上是连续的。对于域$ U \子集M $，我们考虑$ \ Mathcal {f} _U $，$ \ Mathcal {f} $ to $ u $的限制以及相应的正函数$η_U$，通过考虑$ g $的比率和$ \ nathcal ofpoincaré的比例和$ \ nathcal的叶子。首先，我们将$η_U$的变化作为$ u $在linsneto-martins的工作中的Hausdorff感觉中有所不同。其次，Minda证明了双曲线riemann表面$ s $的域Bloch常数的存在，换句话说，这表明，从单位盘到$ S $的每个圆锥形映射到下面的界限上，其扭曲的变形必须在某些均匀半径的双重毛球中本地注入。我们展示了如何推导$ \ Mathcal {f} $的Bloch常数版本
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 $λ_L$ 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 $η_U$ by considering the ratio of $g$ and the Poincaré metric on the leaves of $\mathcal{F}_U$. First, we study the variation of $η_U$ 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 $\mathcal{F}$
