Discrete & Computational Geometry

, Volume 11, Issue 1, pp 35–49 | Cite as

Second derivatives of circle packings and conformal mappings

  • Peter Doyle
  • Zheng-Xu He
  • Burt Rodin
Article

Abstract

William Thurston conjectured that the Riemann mapping functionf from a simply connected region Ω onto the unit disk\(\mathbb{D}\) can be approximated as follows. Almost fill Ω with circles of radius ɛ packed in the regular hexagonal pattern. There is a combinatorially isomorphic packing of circles in\(\mathbb{D}\). The correspondencef ɛ of ɛ-circles in Ω with circles of varying radii in\(\mathbb{D}\) should converge tof after suitable normalization. This was proved in [RS], and in [H] an estimate was obtained which led to an approximation of |f′| in terms off ɛ ; namely, |f′| is the limit of the ratio of the radii of a target circle off ɛ to its source circle. In the present paper we show how to approximatef′ andf″ in terms off ɛ . Explicit rates for the convergence tof, f′, andf″ are obtained. In the special case of convergence to |f′|, the estimate in this paper improves the previously known estimate.

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References

  1. [Ah] D. Aharanov, The hexagonal packing lemma and discrete potential theory,Canad. Math. Bull. 33 (1990), 247–252.MathSciNetCrossRefGoogle Scholar
  2. [H] Z.-X. He, An estimate for hexagonal circle packings,J. Differential Geom.,33 (1991), 395–412.MathSciNetMATHGoogle Scholar
  3. [HR] Z.-X. He and B. Rodin, Convergence of circle packings of finite valence to Riemann mappings, Preprint.Google Scholar
  4. [MR] A. Marden and B. Rodin,On Thurston's Formulation and Proof of Andreev's Theorem, Lecture Notes in Mathematics, Vol. 1435, Springer-Verlag, Berlin, (1990), pp. 103–115.Google Scholar
  5. [R1] B. Rodin, Schwarz's lemma for circle packings, I,Invent. Math.,89 (1987), 271–289.MathSciNetCrossRefMATHGoogle Scholar
  6. [R2] B. Rodin, Schwarz's lemma for circle packings, II,J. Differential Geom.,30 (1989), 539–554.MathSciNetMATHGoogle Scholar
  7. [RS] B. Rodin and D. Sullivan, The convergence of circle packings to the Riemann mapping,J. Differential Geom. 26 (1987), 349–360.MathSciNetMATHGoogle Scholar
  8. [S1] O. Schramm, Rigidity of infinite (circle) packings,J. Amer. Math. Soc.,4 (1991), 127–149.MathSciNetCrossRefGoogle Scholar
  9. [S2] O. Schramm, Existence and uniqueness of packings with specified combinatorics,Israel J. Math. 73 (1991), 321–341.MathSciNetCrossRefGoogle Scholar
  10. [T] W. P. Thurston, The finite Riemann mapping theorem, invited address, International Symposium in Celebration of the Proof of the Bieberbach Conjecture, Purdue University, March 1985.Google Scholar
  11. [W] S.E. Warschawski, On the degree of variation in conformal mapping of variable regions,Trans. Amer. Math. Soc. 69 (1950), 335–356.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1994

Authors and Affiliations

  • Peter Doyle
    • 1
  • Zheng-Xu He
    • 1
  • Burt Rodin
    • 2
  1. 1.Princeton UniversityPrincetonUSA
  2. 2.Department of MathematicsUniversity of California, San DiegoLa JollaUSA

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