The Structure of the Semigroup of Proper Holomorphic Mappings of a Planar Domain to the Unit Disc
The Structure of the Semigroup of Proper Holomorphic Mappings of a Planar Domain to the Unit Disc
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平面域到单位圆盘的真全纯映射半群的结构
DOI：
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发表时间：
2007
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通讯作者：
Faisal Kaleem
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文献类型：
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作者：
S. Bell;Faisal Kaleem
给定由 n 条不相交 Jordan 曲线界定的平面中的有界 n 连通域 Ω，并在每条边界曲线上给定一个点 bj，L. Bieberbach 证明了 Ω 到单位圆盘上存在适当的全纯映射 f，该映射是 n 对一分支覆盖，具有以下属性：f 连续延伸到边界并将每条边界曲线一对一映射到单位圆上，并且 f 将边界上的每个给定点 bj 映射到单位中的点 1圈子。我们将修改 H. Grunsky 对 Bieberbach 结果的证明，以表明存在一个由 2n + 2 个复数变量组成的有理函数，可以生成所有这些图。事实上，我们证明有两个与域相关的 Ahlfors 映射 f1 和 f2，这样任何这样的映射都是通过将右半平面映射到由 c R + i C 组成的单位圆的固定线性分数变换给出的，其中 R 是 2n + 2 函数 $f_1(z)、f_2(z) 和 f_1(b_1) 的有理函数， f_2(b_1),...f_1(b_n),f_2(b_n)$，其中 c 和 C 是满足条件 c > 0 的任意实常数。我们还展示了如何通过有理函数 R 生成到单位圆盘的所有正确全纯映射。
Given a bounded n-connected domain Ω in the plane bounded by n non-intersecting Jordan curves and given one point bj on each boundary curve, L. Bieberbach proved that there exists a proper holomorphic mapping f of Ω onto the unit disc that is an n-to-one branched covering with the properties: f extends continuously to the boundary and maps each boundary curve one-to-one onto the unit circle, and f maps each given point bj on the boundary to the point 1 in the unit circle. We shall modify a proof by H. Grunsky of Bieberbach’s result to show that there is a rational function of 2n + 2 complex variables that generates all of these maps. In fact, we show that there are two Ahlfors maps f1 and f2 associated with the domain such that any such mapping is given by a fixed linear fractional transformation mapping the right half plane to the unit disc composed with c R + i C, where R is a rational function of the 2n + 2 functions $f_1(z),f_2(z), and f_1(b_1), f_2(b_1),...f_1(b_n),f_2(b_n)$, and where c and C are arbitrary real constants subject to the condition c > 0. We also show how to generate all the proper holomorphic mappings to the unit disc via the rational function R.