Rigidity of infinite disk patterns. (English) Zbl 0922.30020

He, Zheng-Xu

Ann. Math. (2) 149, No. 1, 1-33 (1999).

Let \(P\) be a locally finite disk pattern on the complex plane \(\mathbb{C}\) whose combinatorics are described by the one-skelton \(G\) of a triangulation of the open topological disk and whose dihedral angles are equal to a function \(\theta:E\to[0,\pi/2]\) on the set of edges. The author shows that \(P\) is determined up to a euclidean similarity. If \(\theta=0\) this was proved earlier by Rodin and Sullivan and generalized by O. Schramm. Their methods do not work here. The author uses new clever ideas including discrete potential theory, probabilistic methods first developed by K. Stephenson, etc. as well as other important tools.

Reviewer: Dov Aharonov (Haifa)

MSC:

30C25	Covering theorems in conformal mapping theory
30F20	Classification theory of Riemann surfaces
52C25	Rigidity and flexibility of structures (aspects of discrete geometry)
52C26	Circle packings and discrete conformal geometry

Keywords:

packing of the plane; discrete potential theory; maximum principle; extremal length