无限圆盘模式的刚性
Rigidity of infinite disk patterns
摘要：
设P是复平面C上的局部有限圆盘模式，其组合学由开拓扑圆盘的三角剖分的单骨架G描述，其二面角等于边集上的函数Θ：E \to [0，π/2].设P^* 是平面上组合等价的圆盘图案，具有相同的二面角函数。我们证明了P和P^* 之间只有欧几里得相似性的差别。特别地，当二面角函数Θ恒为零时，这产生B的刚性定理。罗丹和D. Sullivan和O. Schramm，他的论点基本上依赖于磁盘内部的成对不相交性。这里的方法是解析的，并使用了最大值原理，顶点极值长度的概念，和递归的一个家庭的电网络，通过放置在边缘上的模式的接触图的电阻。一个类似的刚性属性保持局部有限的磁盘模式在双曲平面，其中的证明遵循一个简单的使用最大值原理。此外，我们有一个统一的结果，磁盘模式。在将来的论文中，本文的技术将扩展到0 \le Θ< π的情况。特别地，我们将证明三维双曲空间中一类无限凸多面体的刚性性质。
英文摘要：
Let P be a locally finite disk pattern on the complex plane C whose combinatorics are described by the one-skeleton G of a triangulation of the open topological disk and whose dihedral angles are equal to a function Θ:E\to [0,π/2] on the set of edges. Let P^* be a combinatorially equivalent disk pattern on the plane with the same dihedral angle function. We show that P and P^* differ only by a euclidean similarity. In particular, when the dihedral angle function Θis identically zero, this yields the rigidity theorems of B. Rodin and D. Sullivan, and of O. Schramm, whose arguments rely essentially on the pairwise disjointness of the interiors of the disks. The approach here is analytical, and uses the maximum principle, the concept of vertex extremal length, and the recurrency of a family of electrical networks obtained by placing resistors on the edges in the contact graph of the pattern. A similar rigidity property holds for locally finite disk patterns in the hyperbolic plane, where the proof follows by a simple use of the maximum principle. Also, we have a uniformization result for disk patterns. In a future paper, the techniques of this paper will be extended to the case when 0 \le Θ< π. In particular, we will show a rigidity property for a class of infinite convex polyhedra in the 3-dimensional hyperbolic space.
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