Acta Mathematica

The C-convergence of hexagonal disk packings to the Riemann map

Zheng-Xu He and Oded Schramm

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The first named author was supported by NSF Grant DMS 96-22068, and the second named author by NSF Grant DMS 94-03548.

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Acta Math., Volume 180, Number 2 (1998), 219-245.

Received: 13 December 1996
First available in Project Euclid: 31 January 2017

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1998 © Institut Mittag-Leffler


He, Zheng-Xu; Schramm, Oded. The C ∞ -convergence of hexagonal disk packings to the Riemann map. Acta Math. 180 (1998), no. 2, 219--245. doi:10.1007/BF02392900.

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  • Aharonov, D., The hexagonal packing lemma and discrete potential theory. Canad. Math. Bull., 33 (1990), 247–252.
  • —, The hexagonal packing lemma and the Rodin Sullivan conjecture. Trans. Amer. Math. Soc., 343 (1994), 157–167.
  • Bowers, P. L., The upper Perron method for labeled complexes with applications to circle packings. Math. Proc. Cambridge Philos. Soc., 114 (1993), 321–345.
  • Colin De Verdière, Y., Un principe variationnel pour les empilements de cercles. Invent. Math., 104 (1991), 655–669.
  • Doyle, P. G., He, Z.-X. & Rodin, B., Second derivatives of circle packings and conformal mappings. Discrete Comput. Geom., 11 (1994), 35–49.
  • He, Z.-X., An estimate for hexagonal circle packings. J. Differential Geom., 33 (1991), 395–412.
  • He, Z.-X., Rigidity of infinite disk patterns. Preprint, 1996.
  • He, Z.-X. & Rodin, B., Convergence of circle packings of finite valence to Riemann mappings. Comm. Anal. Geom., 1 (1993), 31–41.
  • He, Z.-X. & Schramm, O., On the convergence of circle packings to the Riemann map. Invent. Math., 125 (1996), 285–305.
  • Koebe, P., Kontaktprobleme der konformen Abbildung. Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl., 88 (1936), 141–164.
  • Lehto, O., Univalent Functions and Teichmüller Spaces. Graduate Texts in Math., 109. Springer-Verlag, New York, 1987.
  • Marden, A. & Rodin, B., On Thurston's formulation and proof of Andreev's theorem, in Computational Methods and Function Theory (Valparaiso, 1989), pp. 103–115. Lecture Notes in Math., 1435. Springer-Verlag, Berlin, 1990.
  • Rodin, B., Schwarz's lemma for circle packings. Invent. Math., 89 (1987), 271–289.
  • —, Schwarz's lemma for circle packings, II. J. Differential Geom., 30 (1989), 539–554.
  • Rodin, B. & Sullivan, D., The convergence of circle packings to the Riemann mapping. J. Differential Geom., 26 (1987), 349–360.
  • Schramm, O., Circle patterns with the combinatorics of the square grid. Duke Math. J., 86 (1997), 347–389.
  • Stephenson, K., A probablistic proof of Thurston's conjecture on circle packings. Preprint.
  • Thurston, W. P., The Geometry and Topology of 3-manifolds. Princeton Univ. Notes, Princeton, NJ, 1982.
  • Thurston, W. P., The finite Riemann mapping theorem. Unpublished talk given at the International Symposium in Celebration of the Proof of the Bieberbach Conjecture (Purdue University, 1985).