A circle packing measurable Riemann mapping theorem

Williams, G.

doi:10.1090/S0002-9939-06-08200-1

A circle packing measurable Riemann mapping theorem
HTML articles powered by AMS MathViewer

by G. Brock Williams PDF

Proc. Amer. Math. Soc. 134 (2006), 2139-2146 Request permission

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

E. M. Andreev, Convex polyhedra of finite volume in Lobačevskiĭ space, Mat. Sb. (N.S.) 83 (125) (1970), 256–260 (Russian). MR 0273510
Roger W. Barnard and G. Brock Williams, Combinatorial excursions in moduli space, Pacific J. Math. 205 (2002), no. 1, 3–30. MR 1921075, DOI 10.2140/pjm.2002.205.3
Alan F. Beardon and Kenneth Stephenson, The uniformization theorem for circle packings, Indiana Univ. Math. J. 39 (1990), no. 4, 1383–1425. MR 1087197, DOI 10.1512/iumj.1990.39.39062
Christopher Bishop, Conformal welding and Koebe’s theorem, preprint.
Philip L. Bowers and Kenneth Stephenson, A “regular” pentagonal tiling of the plane, Conform. Geom. Dyn. 1 (1997), 58–68. MR 1479069, DOI 10.1090/S1088-4173-97-00014-3
Tomasz Dubejko, Branched circle packings and discrete Blaschke products, Trans. Amer. Math. Soc. 347 (1995), no. 10, 4073–4103. MR 1308008, DOI 10.1090/S0002-9947-1995-1308008-1
Tomasz Dubejko and Kenneth Stephenson, Circle packing: experiments in discrete analytic function theory, Experiment. Math. 4 (1995), no. 4, 307–348. MR 1387696, DOI 10.1080/10586458.1995.10504331
Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmüller theory, Mathematical Surveys and Monographs, vol. 76, American Mathematical Society, Providence, RI, 2000. MR 1730906, DOI 10.1090/surv/076
Zheng-Xu He, Solving Beltrami equations by circle packing, Trans. Amer. Math. Soc. 322 (1990), no. 2, 657–670. MR 974518, DOI 10.1090/S0002-9947-1990-0974518-5
Zheng-Xu He and Burt Rodin, Convergence of circle packings of finite valence to Riemann mappings, Comm. Anal. Geom. 1 (1993), no. 1, 31–41. MR 1230272, DOI 10.4310/CAG.1993.v1.n1.a2
Zheng-Xu He and O. Schramm, Hyperbolic and parabolic packings, Discrete Comput. Geom. 14 (1995), no. 2, 123–149. MR 1331923, DOI 10.1007/BF02570699
Zheng-Xu He and Oded Schramm, On the convergence of circle packings to the Riemann map, Invent. Math. 125 (1996), no. 2, 285–305. MR 1395721, DOI 10.1007/s002220050076
P. Koebe, Kontaktprobleme der Konformen Abbildung, Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 88 (1936), 141–164.
Olli Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR 867407, DOI 10.1007/978-1-4613-8652-0
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
Al Marden and Burt Rodin, On Thurston’s formulation and proof of Andreev’s theorem, Computational methods and function theory (Valparaíso, 1989) Lecture Notes in Math., vol. 1435, Springer, Berlin, 1990, pp. 103–115. MR 1071766, DOI 10.1007/BFb0087901
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
Kenneth Stephenson, A probabilistic proof of Thurston’s conjecture on circle packings, Rend. Sem. Mat. Fis. Milano 66 (1996), 201–291 (1998). MR 1639851, DOI 10.1007/BF02925361
William Thurston, The geometry and topology of $3$-manifolds, Princeton University Notes, preprint.
—, 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.
Masahiko Toki, Moduli of tori obtained by conformal sewing, preprint.
G. Brock Williams, Earthquakes and circle packings, J. Anal. Math. 85 (2001), 371–396. MR 1869616, DOI 10.1007/BF02788088
G. Brock Williams, Noncompact surfaces are packable, J. Anal. Math. 90 (2003), 243–255. MR 2001072, DOI 10.1007/BF02786558
G. Brock Williams, Discrete conformal welding, Indiana Univ. Math. J. 53 (2004), no. 3, 765–804. MR 2086700, DOI 10.1512/iumj.2004.53.2392