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Circle packings in the approximation of conformal mappings. (English) Zbl 0714.30011

Conformal mappings may be approximated by circle packing isomorphisms. This was suggested by Thurston and proved for hexagonal packings by B. Rodin and D. Sullivan [J. Differ. Geom. 26, 349-360 (1987; Zbl 0694.30006)]. Here the author outlines the proof of the corresponding result for non-hexagonal packings. (Details will appear elsewhere.) As in the Rodin-Sullivan proof, it is shown that a certain sequence of affine maps derived from the circle packing isomorphisms is uniformly k- quasiconformal with \(k\to 1\) on compacta, and thus converges to the appropriate conformal mapping. The proof of this “key fact” is quite different here, using methods from the theory of Markov processes.
Reviewer: D.Lesley

MSC:

30C35 General theory of conformal mappings
30G25 Discrete analytic functions
51M99 Real and complex geometry

Keywords:

circle packing

Citations:

Zbl 0694.30006
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Full Text: DOI

References:

[1] Alan F. Beardon and Kenneth Stephenson, The Schwarz-Pick lemma for circle packings, Illinois J. Math. 35 (1991), no. 4, 577 – 606. Alan F. Beardon and Kenneth Stephenson, Addendum to our paper: ”The Schwarz-Pick lemma for circle packings”, Illinois J. Math. 36 (1992), no. 2, 177. · Zbl 0753.30016
[2] Peter G. Doyle and J. Laurie Snell, Random walks and electric networks, Carus Mathematical Monographs, vol. 22, Mathematical Association of America, Washington, DC, 1984. · Zbl 0583.60065
[3] Burt Rodin and Dennis Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987), no. 2, 349 – 360. · Zbl 0694.30006
[4] W. Thurston, The geometry and topology of 3-manifolds, preprint, Princeton Univ. Notes.
[5] W. Thurston, The finite Riemann mapping theorem, Invited talk, An International Symposium at Purdue University on the occasion of the proof of the Bieberbach conjecture, March 1985.
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