Rigidity of infinite disk patterns

Let P be a locally finite disk pattern on the complex plane C whose combina torics are described by the one-skeleton G of a triangulation of the open t opological disk and whose dihedral angles are equal to a function Theta E - -> [0, pi/2] on the set of edges. Let P* be a combinatorially equivalent di sk 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 Theta is identically zero, this yields the rigidity theorems of B. Rodin and D. Sullivan, and of O. Sc hramm, whose arguments rely essentially on the pairwise disjointness of the interiors of the disks. The approach here is analytical, and uses the maxi mum 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 h yperbolic plane, where the proof follows by a simple use of the maximum pri nciple. Also, we have a uniformization result for disk patterns. In a future paper, the techniques of this paper will be extended to the cas e when 0 less than or equal to Theta < pi. In particular, we will show a ri gidity property for a class of infinite convex polyhedra in the 3-dimension al hyperbolic space.