Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A circle packing measurable Riemann mapping theorem


Author: G. Brock Williams
Journal: Proc. Amer. Math. Soc. 134 (2006), 2139-2146
MSC (2000): Primary 52C26, 30F60
DOI: https://doi.org/10.1090/S0002-9939-06-08200-1
Published electronically: January 4, 2006
MathSciNet review: 2215785
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a circle packing version of the Measurable Riemann Mapping Theorem in the spirit of Rodin and Sullivan's Circle Packing Riemann Mapping Theorem. We also construct circle packing maps of the plane onto itself with prescribed dilatation.


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

  • 1. E. M. Andreev, Convex polyhedra of finite volume in Lobacevskii space, Math. USSR Sbornik 12 (1970), 255-259 (English). MR 0273510 (42:8388)
  • 2. R. W. Barnard and G. Brock Williams, Combinatorial excursions in moduli space, Pacific J. Math. 205 (2002), no. 1, 3-30. MR 1921075 (2003f:52017)
  • 3. Alan F. Beardon and Kenneth Stephenson, The uniformization theorem for circle packings, Indiana Univ. Math. J. 39 (1990), 1383-1425. MR 1087197 (92b:52038)
  • 4. Christopher Bishop, Conformal welding and Koebe's theorem, preprint.
  • 5. Philip L. Bowers and Kenneth Stephenson, A regular pentagonal tiling of the plane, Conform. Geom. Dyn. 1 (1997), 58-68. MR 1479069 (99d:52016)
  • 6. Tomasz Dubejko, Branched circle packings and discrete Blaschke products, Trans. Amer. Math. Soc. 347 (1995), no. 10, 4073-4103. MR 1308008 (95m:30045)
  • 7. Tomasz Dubejko and Kenneth Stephenson, Circle packing: Experiments in discrete analytic function theory, Experiment. Math. 4 (1995), no. 4, 307-348. MR 1387696 (97f:52027)
  • 8. Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmüller theory, Mathematical Surveys and Monographs, vol. 76, American Mathematical Society, 2000. MR 1730906 (2001d:32016)
  • 9. Zheng-Xu He, Solving Beltrami equations by circle packing, Trans. Amer. Math. Soc. 322 (1990), 657-670. MR 0974518 (91c:30032)
  • 10. Zheng-Xu He and Burt Rodin, Convergence of circle packings of finite valence to Riemann mappings, Comm. in Analysis and Geometry 1 (1993), 31-41. MR 1230272 (94m:30019)
  • 11. Zheng-Xu He and Oded Schramm, Hyperbolic and parabolic packings, Discrete & Computational Geom. 14 (1995), 123-149. MR 1331923 (96h:52017)
  • 12. -, On the convergence of circle packings to the Riemann map, Invent. Math. 125 (1996), 285-305. MR 1395721 (97i:30009)
  • 13. P. Koebe, Kontaktprobleme der Konformen Abbildung, Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 88 (1936), 141-164.
  • 14. O. Lehto, Univalent functions and Teichmüller spaces, Springer-Verlag, Berlin, Heidelberg, New York, 1987. MR 0867407 (88f:30073)
  • 15. O. Lehto and K.I. Virtanen, Quasiconformal mappings in the plane, second ed., Springer-Verlag, Berlin, Heidelberg, New York, 1973. MR 0344463 (49:9202)
  • 16. Al Marden and Burt Rodin, On Thurston's formulation and proof of Andreev's theorem, Computational Methods and Function Theory, Proceeding, Valparaiso 1989, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1990, Lecture Notes in Mathematics, Vol. 1435, pp. 103-115. MR 1071766 (92b:52040)
  • 17. Burt Rodin and Dennis Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geometry 26 (1987), 349-360. MR 0906396 (90c:30007)
  • 18. Kenneth Stephenson, A probabilistic proof of Thurston's conjecture on circle packings, Rend. Sem. Mat. Fis. Milano 66 (1996), 201-291. MR 1639851 (99m:52024)
  • 19. William Thurston, The geometry and topology of $ 3$-manifolds, Princeton University Notes, preprint.
  • 20. -, 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.
  • 21. Masahiko Toki, Moduli of tori obtained by conformal sewing, preprint.
  • 22. G. Brock Williams, Earthquakes and circle packings, J. Anal. Math. 85 (2001), 371-396. MR 1869616 (2003e:57032)
  • 23. -, Noncompact surfaces are packable, J. Anal. Math. 90 (2003). MR 2001072 (2004h:30055)
  • 24. -, Discrete conformal welding, Indiana Univ. Math. J. 53 (2004), no. 3, 765-804. MR 2086700 (2005f:30018)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 52C26, 30F60

Retrieve articles in all journals with MSC (2000): 52C26, 30F60


Additional Information

G. Brock Williams
Affiliation: Department of Mathematics, Texas Tech University, Lubbock, Texas 79409
Email: williams@math.ttu.edu

DOI: https://doi.org/10.1090/S0002-9939-06-08200-1
Keywords: Circle packing, quasiconformal maps, Teichm\"uller theory
Received by editor(s): December 4, 2002
Received by editor(s) in revised form: February 10, 2005
Published electronically: January 4, 2006
Additional Notes: The author gratefully acknowledges the support of the Texas Tech University Research Enhancement Fund.
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society