Rigidity of Infinite Disk Patterns

Let P be a locally finite disk pattern on the complex plane C whose combinatorics is 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 → [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 Θ is 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 ≤ Θ < π . In particular, we will show a rigidity property for a class of infinite convex polyhedra in the 3-dimensional hyperbolic space.