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Convergence of circle packings of finite valence to Riemann mappings. (English) Zbl 0777.30003

Summary: In a paper of B. Rodin and D. Sullivan, the conjecture by W. Thurston that the hexagonal circle packings can be used to approximate the Riemann mapping (in the topology of uniform convergence in compact subsets) is proved; and in 1989, Z. X. He has shown that the derivatives of these approximations are convergent. We show in Section 1 that the methods used by Rodin and Sullivan in the case of hexagonal packings can be easily extended to the case of nonhexagonal circle packing with bounded radii ratios. We note that Stephenson had taken the major steps toward such an extension. Although he follows the overall strategy of Rodin and Sullivan, he replaces certain key steps by parabolistic arguments which have an interesting interpretation in terms of the flow of electricity in a network. In Section 2, we show that the method of Z. X. He can be extended to a more general class of non hexagonal packings. Specifically, the restriction that the radii ratios be bounded can be replaced by the much weaker condition that the circle packings have uniformly bounded valence.

MSC:

30C20 Conformal mappings of special domains

Keywords:

circle packings
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