The hexagonal packing lemma and Rodin Sullivan conjecture

Aharonov, Dov

doi:10.1090/S0002-9947-1994-1162100-2

The hexagonal packing lemma and Rodin Sullivan conjecture
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Trans. Amer. Math. Soc. 343 (1994), 157-167 Request permission

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

Dov Aharonov, The hexagonal packing lemma and discrete potential theory, Canad. Math. Bull. 33 (1990), no. 2, 247–252. MR 1060381, DOI 10.4153/CMB-1990-039-5
Zheng-Xu He, An estimate for hexagonal circle packings, J. Differential Geom. 33 (1991), no. 2, 395–412. MR 1094463
O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 126, Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas. MR 0344463, DOI 10.1007/978-3-642-65513-5
Burt Rodin, Schwarz’s lemma for circle packings, Invent. Math. 89 (1987), no. 2, 271–289. MR 894380, DOI 10.1007/BF01389079
Burt Rodin, Schwarz’s lemma for circle packings. II, J. Differential Geom. 30 (1989), no. 2, 539–554. MR 1010171
Burt Rodin and Dennis Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987), no. 2, 349–360. MR 906396