共形映射的圆模式逼近
Approximation of conformal mappings by circle patterns
摘要：
圆模式是平面中的圆的配置，其组合学由可平面图G给出，使得G的每个顶点对应于一个圆。如果两个顶点由G中的边连接，则相应的圆以$(0,π)$中的交角相交。用两个圆模式序列逼近给定的共形映射$g$及其一阶导数。对于$g$的域，我们使用嵌入的圆模式，其中所有圆具有相同的半径减小到0，并且具有均匀有界的相交角。图像圆图案具有相同的组合和交叉角，并且根据$g'$（$|g'|$或$\arg g'$）的值从边界条件（半径或角度）确定。对于拟对称圆模式，收敛结果被加强为紧子集上的$C^\infty$ -收敛.
英文摘要：
A circle pattern is a configuration of circles in the plane whose combinatorics is given by a planar graph G such that to each vertex of G corresponds a circle. If two vertices are connected by an edge in G, the corresponding circles intersect with an intersection angle in $(0,π)$. Two sequences of circle patterns are employed to approximate a given conformal map $g$ and its first derivative. For the domain of $g$ we use embedded circle patterns where all circles have the same radius decreasing to 0 and which have uniformly bounded intersection angles. The image circle patterns have the same combinatorics and intersection angles and are determined from boundary conditions (radii or angles) according to the values of $g'$ ($|g'|$ or $\arg g'$). For quasicrystallic circle patterns the convergence result is strengthened to $C^\infty$-convergence on compact subsets.
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