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Circle packings and conformal representations: A new proof of the Rodin-Sullivan theorem. (Empilements de cercles et représentations conformes: Une nouvelle preuve du théorème de Rodin-Sullivan.) (French) Zbl 0978.30003

In the paper under review the author supplies a new proof of the well-known result of B. Rodin and D. P. Sullivan [J. Differ. Geom. 26, No. 2, 349-360 (1987; Zbl 0694.30006)], to the effect that a certain iterative procedure involving circle packings converges to give a solution to the Riemann mapping problem. This had been previously noted without proof by W. Thurston. Each step in the iteration can be viewed as a solution of a discrete form of the Riemann mapping problem. The work of Rodin and Sullivan gave rise to a considerable industry devoted to circle packing and related issues in discrete potential theory, much of which seems to be centered around producing discrete analogues to the classical “continuous” geometric function theory. In a way, a lot of the work in this field has (or, at least, should have) its roots in the pionerring work of R. J. Duffin in the 1950’s.
The proof of Rodin and Sullivan, while containing some nice geometric ideas, was found by some to be wanting, since at a crucial point in the argument, where they needed to show rigidity of the hexagonal circle packing, Rodin and Sullivan appealed to a deep result of A. Marden on infinitely generated Schottky groups, which seemed out of character with the elementary nature of the rest of the argument. This has since been remedied by a number of the authors.
In the paper under review the author develops discrete analogues of harmonic functions and of the Schrödinger equation, including a discrete analogue of the classical Harnack inequality for harmonic functions, and uses these to give an elegant self-contained proof of the theorem of Rodin and Sullivan.

MSC:

30C30 Schwarz-Christoffel-type mappings
52C15 Packing and covering in \(2\) dimensions (aspects of discrete geometry)
65E05 General theory of numerical methods in complex analysis (potential theory, etc.)

Citations:

Zbl 0694.30006
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