Noncompact surfaces are packable

  • G. Brock Williams
Article
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Abstract

We show that every noncompact Riemann surface of finite type supports a circle packing. This extends earlier work of Robert Brooks [6] and Phil Bowers and Ken Stephenson [3, 4], who showed that the packable surfaces are dense in moduli space.

Keywords

Riemann Surface Fundamental Domain Finite Type Circle Packing Closed Chain 
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References

  1. [1]
    W. Barnard and G. B. Williams,Combinatorial excursions in moduli space, Pacific J. Math.205 (2002), 3–30.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    A. F. Beardon and K. Stephenson,The uniformization theorem for circle packings, Indiana Univ. Math. J.39 (1990), 1383–1425.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    P. L. Bowers and K. Stephenson,The set of circle packing points in the Teichmüller space of a surface of finite conformai type is dense, Math. Proc. Camb. Phil. Soc.111 (1992), 487–513.MATHMathSciNetGoogle Scholar
  4. [4]
    P. L. Bowers and K. Stephenson,Circle packings in surfaces of finite type: An in situ approach with application to moduli, Topology32 (1993), 157–183.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    R. Brooks,On the deformation theory of classical Schottky groups, Duke Math. J.52 (1985), 1009–1024.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    R. Brooks,Circles packings and co-compact extensions of Kleinian groups, Invent. Math.86 (1986), 461–469.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    T. Dubejko and K. Stephenson,Circle packing: Experiments in discrete analytic function theory, Experiment. Math.4 (1995), 307–348.MATHMathSciNetGoogle Scholar
  8. [8]
    Zheng-Xu He and O. Schramm,Fixed points, Koebe uniformization and circle packings, Ann. of Math.137 (1993), 369–406.CrossRefMathSciNetGoogle Scholar
  9. [9]
    Zheng-Xu He and O. Schramm,Hyperbolic and parabolic packings, Discrete Comput. Geom.14 (1995), 123–149.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Zheng-Xu He and O. Schramm,On the convergence of circle packings to the Riemann map, Invent. Math.125 (1996), 285–305.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    B. Rodin and D. Sullivan,The convergence of circle packings to the Riemann mapping, J. Differential Geom.26 (1987), 349–360.MATHMathSciNetGoogle Scholar
  12. [12]
    K. Stephenson, Course Notes for Seminar in Analysis, Chapter 10, Fall 2001, http://www.math.utk,edu/kens/cp01/.Google Scholar
  13. [13]
    W. Thurston,The finite Riemann mapping theorem, 1985, Invited talk, An International Symposium at Purdue University on the occasion of the proof of the Bieberbach conjecture, March 1985.Google Scholar
  14. [14]
    G. B. Williams,Earthquakes and circle packings, J. Analyse Math.85 (2001), 371–396.MATHCrossRefGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2003

Authors and Affiliations

  • G. Brock Williams
    • 1
  1. 1.Department of MathematicsTexas Tech University LUBBOCKUSA

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