Finite simultaneous bending

Finite simultaneous bending
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Conform. Geom. Dyn. 10 (2006), 203-226

DOI: https://doi.org/10.1090/S1088-4173-06-00119-6

Published electronically: September 21, 2006

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We construct a finite approximation to a Jordan curve with the given pair of bending measured laminations.

References

Lipman Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960), 94–97. MR 111834, DOI 10.1090/S0002-9904-1960-10413-2
Francis Bonahon, Kleinian groups which are almost Fuchsian, J. Reine Angew. Math. 587 (2005), 1–15. MR 2186972, DOI 10.1515/crll.2005.2005.587.1
Xiliang Bao and Francis Bonahon, Hyperideal polyhedra in hyperbolic 3-space, Bull. Soc. Math. France 130 (2002), no. 3, 457–491 (English, with English and French summaries). MR 1943885, DOI 10.24033/bsmf.2426
Francis Bonahon and Jean-Pierre Otal, Laminations measurées de plissage des variétés hyperboliques de dimension 3, Ann. of Math. (2) 160 (2004), no. 3, 1013–1055 (French, with English summary). MR 2144972, DOI 10.4007/annals.2004.160.1013
Phil Bowers and Kenneth Stephenson, A branched Andreev-Thurston theorem for circle packings of the sphere, Proc. London Math. Soc. (3) 73 (1996), no. 1, 185–215. MR 1387087, DOI 10.1112/plms/s3-73.1.185

Lengths are coordinates for convex structures

D. B. A. Epstein and A. Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 113–253. MR 903852
F. P. Gardiner, J. Hu, and N. Lakic, Earthquake curves, Complex manifolds and hyperbolic geometry (Guanajuato, 2001) Contemp. Math., vol. 311, Amer. Math. Soc., Providence, RI, 2002, pp. 141–195. MR 1940169, DOI 10.1090/conm/311/05452
Zheng-Xu He, Rigidity of infinite disk patterns, Ann. of Math. (2) 149 (1999), no. 1, 1–33. MR 1680531, DOI 10.2307/121018
Craig D. Hodgson and Igor Rivin, A characterization of compact convex polyhedra in hyperbolic $3$-space, Invent. Math. 111 (1993), no. 1, 77–111. MR 1193599, DOI 10.1007/BF01231281
Ravi S. Kulkarni and Ulrich Pinkall, A canonical metric for Möbius structures and its applications, Math. Z. 216 (1994), no. 1, 89–129. MR 1273468, DOI 10.1007/BF02572311
Linda Keen and Caroline Series, Pleating invariants for punctured torus groups, Topology 43 (2004), no. 2, 447–491. MR 2052972, DOI 10.1016/S0040-9383(03)00052-1
Linda Keen and Caroline Series, How to bend pairs of punctured tori, Lipa’s legacy (New York, 1995) Contemp. Math., vol. 211, Amer. Math. Soc., Providence, RI, 1997, pp. 359–387. MR 1476997, DOI 10.1090/conm/211/02830

On geometry of convex polyhedra in hyperbolic $3$-space

Igor Rivin, A characterization of ideal polyhedra in hyperbolic $3$-space, Ann. of Math. (2) 143 (1996), no. 1, 51–70. MR 1370757, DOI 10.2307/2118652
Igor Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. of Math. (2) 139 (1994), no. 3, 553–580. MR 1283870, DOI 10.2307/2118572
Caroline Series, On Kerckhoff minima and pleating loci for quasi-Fuchsian groups, Geom. Dedicata 88 (2001), no. 1-3, 211–237. MR 1877217, DOI 10.1023/A:1013171204254

Limits of quasifuchsian groups with small bending

Harumi Tanigawa, Grafting, harmonic maps and projective structures on surfaces, J. Differential Geom. 47 (1997), no. 3, 399–419. MR 1617652

The geometry and topology of $3$-manifolds

Earthquakes in two-dimensional hyperbolic geometry