On the Teichmüller theory of circle patterns

He, Zhengxu; Liu, Jinsong

doi:10.1090/S0002-9947-2013-05892-8

On the Teichmüller theory of circle patterns
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by Zhengxu He and Jinsong Liu PDF

Trans. Amer. Math. Soc. 365 (2013), 6517-6541 Request permission

Abstract:

Given a circle pattern on the Riemann sphere $\hat {\mathbb {C}}$, in this paper we prove that its quasiconformal deformation space can be naturally identified with the product of the Teichmüller spaces of its interstices.

References

Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
Alan F. Beardon and Kenneth Stephenson, The Schwarz-Pick lemma for circle packings, Illinois J. Math. 35 (1991), no. 4, 577–606. MR 1115988
Robert Brooks, On the deformation theory of classical Schottky groups, Duke Math. J. 52 (1985), no. 4, 1009–1024. MR 816397, DOI 10.1215/S0012-7094-85-05253-6
Peter Doyle, Zheng-Xu He, and Burt Rodin, Second derivatives of circle packings and conformal mappings, Discrete Comput. Geom. 11 (1994), no. 1, 35–49. MR 1244888, DOI 10.1007/BF02573993
D. Gilbarg $\&$ N. S. Trudinger, Elliptic differential equations of second order (Springer-Verlag, Berlin, 1998).
Zheng-Xu He, Rigidity of infinite disk patterns, Ann. of Math. (2) 149 (1999), no. 1, 1–33. MR 1680531, DOI 10.2307/121018
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
Zheng-Xu He and Oded Schramm, The $C^\infty$-convergence of hexagonal disk packings to the Riemann map, Acta Math. 180 (1998), no. 2, 219–245. MR 1638772, DOI 10.1007/BF02392900
Craig D. Hodgson, Deduction of Andreev’s theorem from Rivin’s characterization of convex hyperbolic polyhedra, Topology ’90 (Columbus, OH, 1990) Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 185–193. MR 1184410
Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors. MR 1215481, DOI 10.1007/978-4-431-68174-8
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
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
I. A. Rivin. $\&$ C. Hodgson, ‘A characterization of compact convex polyhedra in hyperbolic 3-space’, Invent. Math., 111 (1993) 77-111.
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
M. Schiffer and N. S. Hawley, Connections and conformal mapping, Acta Math. 107 (1962), 175–274. MR 182720, DOI 10.1007/BF02545790
Oded Schramm, Rigidity of infinite (circle) packings, J. Amer. Math. Soc. 4 (1991), no. 1, 127–149. MR 1076089, DOI 10.1090/S0894-0347-1991-1076089-9
C. L. Siegel., Topics in complex function theory, Volume.1. (Wiley-Interscience, 1969).
Kenneth Stephenson, Circle packings in the approximation of conformal mappings, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 407–415. MR 1049434, DOI 10.1090/S0273-0979-1990-15946-4
W. Thurston, The geometry and topology of three manifolds, (Princeton University Press, 1977).