Rigidity of infinite disk patterns

Zheng-Xu He

Review from Zentralblatt MATH:

Let $P$ be a locally finite disk pattern on the complex plane $\bbfC$ 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.

Reviewed by Dov Aharonov

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

Classification (MSC2000): 30C99

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