无限有边界圆填充的存在性与唯一性定理

无限无边界的单连通复形有两种基本类型,即双曲型和抛物型,其对应的圆填充分别填满双曲平面和欧式平面。主要讨论无限有边界的单连通复形K的情形,证明了在双曲平面内存在一个关于K的单叶圆填充P,在P中与K的边界顶点对应的圆是极限圆;这个圆填充P在允许其极限圆与单位圆周存在空隙的意义下是完备的;并且P对于单位圆盘D的M bius变换来说是唯一的。

The Existence and Uniqueness Theorem of Infinite Circle Packings with Boundary

It is known that the hyperbolic and parabolic complex are two fundamental types for infinite and simply connected complexes, whose corresponding circle packings fill the hyperbolic and the Euclidean plane, respectively. Given an infinite simply connected complex K with boundary, it is proved that there exists an univalent circle packing P for K in the hyperbolic plane D whose circles associate with boundary vertices of K are horocycles, which is complete in the sense of permitting the existence of interstices between horocycles and unit circle D. Moreover, the circle packing P is unique up to Mbius transformations of D.