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Transactions of the American Mathematical Society

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The hexagonal packing lemma and Rodin Sullivan conjecture


Author: Dov Aharonov
Journal: Trans. Amer. Math. Soc. 343 (1994), 157-167
MSC: Primary 30C85; Secondary 30C62, 52C15
DOI: https://doi.org/10.1090/S0002-9947-1994-1162100-2
MathSciNet review: 1162100
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Abstract: The Hexagonal Packing Lemma of Rodin and Sullivan [6] states that $ {s_n} \to 0$ as $ n \to \infty $. Rodin and Sullivan conjectured that $ {s_n} = O(1/n)$. This has been proved by Z-Xu He [2]. Earlier, the present author proved the conjecture under some additional restrictions [1].

In the following we are able to remove these restrictions, and thus give an alternative proof of the RS conjecture. The proof is based on our previous article [1]. It is completely different from the proof of He, and it is mainly based on discrete potential theory, as developed by Rodin for the hexagonal case [4].


References [Enhancements On Off] (What's this?)

  • [1] D. Aharonov, The hexagonal packing lemma and discrete potential theory, Canad. Math. Bull. 33 (1990), 277-252. MR 1060381 (91h:31011)
  • [2] Z-Xu He, An estimate for hexagonal circle packings, J. Differential Geom. 33 (1991), 395-412. MR 1094463 (92b:52039)
  • [3] O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, Springer-Verlag, New York, Heidelberg, and Berlin, 1973, 258 pp. MR 0344463 (49:9202)
  • [4] B. Rodin, Schwartz's Lemma for circle packings, Invent. Math. 89 (1987), 271-289. MR 894380 (88h:11043)
  • [5] -, Schwartz's Lemma for circle packings. II, J. Differential Geom. 30 (1989), 539-554. MR 1010171 (90m:11099)
  • [6] B. Rodin and D. Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987), 349-360. MR 906396 (90c:30007)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1162100-2
Article copyright: © Copyright 1994 American Mathematical Society

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