Abstract:We construct a finite approximation to a Jordan curve with the given pair of bending measured laminations.
- 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 CS Y.E. Choi, C. Series, Lengths are coordinates for convex structures, J. Differential Geometry, to appear.
- 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 R1 I. Rivin, On geometry of convex polyhedra in hyperbolic $3$-space, Ph.D. Thesis, Princeton, 1986.
- 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 S2 C. Series, Limits of quasifuchsian groups with small bending, Duke J. Mathematics, to appear.
- Harumi Tanigawa, Grafting, harmonic maps and projective structures on surfaces, J. Differential Geom. 47 (1997), no. 3, 399–419. MR 1617652 Th1 W.P. Thurston, The geometry and topology of $3$-manifolds, Princeton University Lecture Notes, Online at http://www.msri.org/publications/books/gt3m, 1982. Th2 W.P. Thurston, Earthquakes in two-dimensional hyperbolic geometry, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984), 91–112, London Math. Soc. Lecture Note Ser., 112, Cambridge Univ. Press, Cambridge, 1986.
- Reza Chamanara
- Affiliation: Institute for Studies in Theoretical Physics and Mathematics (IMP), Tehran, Iran
- Address at time of publication: Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794-3660
- Email: email@example.com
- Received by editor(s): March 22, 2004
- Received by editor(s) in revised form: April 3, 2006
- Published electronically: September 21, 2006
- Additional Notes: This research was in part supported by a grant from IPM (No. 83510120)
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
- Journal: Conform. Geom. Dyn. 10 (2006), 203-226
- MSC (2000): Primary 51M15, 51B10; Secondary 51N25, 51M10, 30F40
- DOI: https://doi.org/10.1090/S1088-4173-06-00119-6
- MathSciNet review: 2261049