Geometry & Topology Volume 19, issue 4 (2015)

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ISSN (electronic): 1364-0380

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Discrete conformal maps and ideal hyperbolic polyhedra

Alexander I Bobenko, Ulrich Pinkall and Boris A Springborn

Geometry & Topology 19 (2015) 2155–2215

DOI: 10.2140/gt.2015.19.2155

Bibliography

1	A D Alexandrov, Convex polyhedra, Springer (2005) MR2127379
2	E M Andreev, On convex polyhedra in Lobačevski\ui spaces, Math. USSR-Sb. 10 (1970) 413
3	E M Andreev, On convex polyhedra of finite volume in Lobačevski\ui space, Math. USSR-Sb. 12 (1970) 255
4	A I Bobenko, T Hoffmann, B A Springborn, Minimal surfaces from circle patterns: geometry from combinatorics, Ann. of Math. 164 (2006) 231 MR2233848
5	A I Bobenko, I Izmestiev, Alexandrov's theorem, weighted Delaunay triangulations and mixed volumes, Ann. Inst. Fourier (Grenoble) 58 (2008) 447 MR2410380
6	A I Bobenko, B A Springborn, Variational principles for circle patterns and Koebe's theorem, Trans. Amer. Math. Soc. 356 (2004) 659 MR2022715
7	B Chow, F Luo, Combinatorial Ricci flows on surfaces, J. Differential Geom. 63 (2003) 97 MR2015261
8	Y Colin de Verdière, Un principe variationnel pour les empilements de cercles, Invent. Math. 104 (1991) 655 MR1106755
9	J Dai, X D Gu, F Luo, Variational principles for discrete surfaces, Adv. Lect. Math. 4, International Press (2008) MR2439807
10	R J Duffin, Distributed and lumped networks, J. Math. Mech. 8 (1959) 793 MR0106032
11	D B A Epstein, R C Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988) 67 MR918457
12	V V Fock, Dual Teichmüller spaces, arXiv:dg-ga/9702018
13	D Futer, F Guéritaud, From angled triangulations to hyperbolic structures, from: "Interactions between hyperbolic geometry, quantum topology and number theory" (editors A Champanerkar, O Dasbach, E Kalfagianni, I Kofman, W Neumann, N Stoltzfus), Contemp. Math. 541, Amer. Math. Soc. (2011) 159 MR2796632
14	M von Gagern, J Richter-Gebert, Hyperbolization of Euclidean ornaments, Electron. J. Combin. 16 (2009) MR2515775
15	I M Gelfand, M M Kapranov, A V Zelevinsky, Discriminants, resultants and multidimensional determinants, Birkhäuser (1994) MR1264417
16	X D Gu, S T Yau, Computational conformal geometry, Adv. Lect. Math. 3, International Press (2008) MR2439718
17	Z X He, O Schramm, Fixed points, Koebe uniformization and circle packings, Ann. of Math. 137 (1993) 369 MR1207210
18	R Kenyon, Lectures on dimers, from: "Statistical mechanics" (editors S Sheffield, T Spencer), IAS/Park City Math. Ser. 16, Amer. Math. Soc. (2009) 191 MR2523460
19	R Kenyon, A Okounkov, Planar dimers and Harnack curves, Duke Math. J. 131 (2006) 499 MR2219249
20	R Kenyon, A Okounkov, S Sheffield, Dimers and amoebae, Ann. of Math. 163 (2006) 1019 MR2215138
21	P Koebe, Kontaktprobleme der konformen Abbildung, Ber. Sächs. Akad. Wiss. Leipzig, Math.-phys. Kl. 88 (1936) 141
22	G Leibon, Characterizing the Delaunay decompositions of compact hyperbolic surfaces, Geom. Topol. 6 (2002) 361 MR1914573
23	L Lewin, Polylogarithms and associated functions, North-Holland (1981) MR618278
24	F Luo, Combinatorial Yamabe flow on surfaces, Commun. Contemp. Math. 6 (2004) 765 MR2100762
25	F Luo, Rigidity of polyhedral surfaces, from: "Fourth International Congress of Chinese Mathematicians" (editors L Ji, K Liu, L Yang, S T Yau), AMS/IP Stud. Adv. Math. 48, Amer. Math. Soc. (2010) 201 MR2744222
26	G Mikhalkin, Amoebas of algebraic varieties and tropical geometry, from: "Different faces of geometry" (editors S Donaldson, Y Eliashberg, M Gromov), Int. Math. Ser. 3, Kluwer/Plenum (2004) 257 MR2102998
27	J Milnor, Hyperbolic geometry: The first 150 years, Bull. Amer. Math. Soc. 6 (1982) 9 MR634431
28	J Milnor, How to compute volume in hyperbolic space, from: "Collected papers, Vol. $1$", Publish or Perish (1994) 189 MR1277810
29	J Milnor, The Schläfli differential equality, from: "Collected papers, Vol. $1$", Publish or Perish (1994) 281 MR1277810
30	A Papadopoulos, editor, Handbook of Teichmüller theory, Vol. I, IRMA Lectures Math. Theor. Physics 11, Euro. Math. Soc. (2007) MR2284826
31	M Passare, H Rullgård, Amoebas, Monge–Ampère measures and triangulations of the Newton polytope, Duke Math. J. 121 (2004) 481 MR2040284
32	R C Penner, The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987) 299 MR919235
33	U Pinkall, K Polthier, Computing discrete minimal surfaces and their conjugates, Experiment. Math. 2 (1993) 15 MR1246481
34	I Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. of Math. 139 (1994) 553 MR1283870
35	I Rivin, Intrinsic geometry of convex ideal polyhedra in hyperbolic $3$–space, from: "Analysis, algebra, and computers in mathematical research" (editor M Gyllenberg), Lecture Notes Pure Appl. Math. 156, Dekker (1994) 275 MR1280952
36	I Rivin, Combinatorial optimization in geometry, Adv. in Appl. Math. 31 (2003) 242 MR1985831
37	M Roček, R M Williams, The quantization of Regge calculus, Z. Phys. C 21 (1984) 371 MR734698
38	B Rodin, D Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987) 349 MR906396
39	R K W Roeder, J H Hubbard, W D Dunbar, Andreev's theorem on hyperbolic polyhedra, Ann. Inst. Fourier (Grenoble) 57 (2007) 825 MR2336832
40	B Springborn, P Schröder, U Pinkall, Conformal equivalence of triangle meshes, ACM Trans. Graph. 27 (2008) 11
41	K Stephenson, Introduction to circle packing, Cambridge Univ. Press (2005) MR2131318
42	W P Thurston, The geometry and topology of three-manifolds, lecture notes (1979)
43	W P Thurston, Minimal stretch maps between hyperbolic surfaces, (1998) arXiv:math/9801039
44	M Troyanov, Les surfaces euclidiennes à singularités coniques, Enseign. Math. 32 (1986) 79 MR850552