具有无界度的无限圆填充的刚性(英文)

Rigidity of Infinite Circle Packings of Unbounded Degree

刚性是圆填充理论的一个重要的性质.已经知道,平面上无限的有界度的圆填充的刚性可以用环绕数的方法来证明.本文应用环绕数和指标的技术,结合有限覆盖定理证明了几乎填满整个黎曼球面具有相同复形的无限无界度圆填充对M(o|¨)bius变换来说是等价的,也就是,一个圆填充是另一个圆填充在M(o|¨)bius变换下的像.这给出了无限无界度圆填充的刚性的一种新的证明.

Rigidity is an important property in the theory of circle packings. It is known that rigidity of infinite circle packings of bounded degree can be proved by using the method of winding number. In＇this paper using the methods of winding number and index, combining with the finite covering theorem, we show that any two infinite circle packings of unbounded degree, which have the same combinatorics and almost fill the whole sphere, are MSbius equivalent, that is, one is the image of another under M6bius transformations. This provides a new proof of the rigidity of infinite circle packings with unbounded degree.