Geometry & Topology Volume 26, issue 3 (2022)

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Discrete conformal geometry of polyhedral surfaces and its convergence

Feng Luo, Jian Sun and Tianqi Wu

Geometry & Topology 26 (2022) 937–987

DOI: 10.2140/gt.2022.26.937

Bibliography

1	L V Ahlfors, Lectures on quasiconformal mappings, 38, Amer. Math. Soc. (2006) MR2241787
2	A D Alexandrov, Selected works, II : Intrinsic geometry of convex surfaces, Chapman & Hall (2006) MR2193913
3	A I Bobenko, U Pinkall, B A Springborn, Discrete conformal maps and ideal hyperbolic polyhedra, Geom. Topol. 19 (2015) 2155 MR3375525
4	U Bücking, Approximation of conformal mappings by circle patterns, Geom. Dedicata 137 (2008) 163 MR2449151
5	U Bücking, Approximation of conformal mappings using conformally equivalent triangular lattices, from: "Advances in discrete differential geometry" (editor A I Bobenko), Springer (2016) 133 MR3587184
6	F R K Chung, Spectral graph theory, 92, Amer. Math. Soc. (1997) MR1421568
7	S Dai, H Ge, S Ma, Rigidity of the hexagonal Delaunay triangulated plane, Peking Math. J. 5 (2022) 1
8	D B A Epstein, A Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, from: "Analytical and geometric aspects of hyperbolic space" (editor D B A Epstein), Lond. Math. Soc. Lect. Note Ser. 111, Cambridge Univ. Press (1987) 113 MR903852
9	D Glickenstein, Discrete conformal variations and scalar curvature on piecewise flat two- and three-dimensional manifolds, J. Differential Geom. 87 (2011) 201 MR2788656
10	X D Gu, F Luo, J Sun, T Wu, A discrete uniformization theorem for polyhedral surfaces, J. Differential Geom. 109 (2018) 223 MR3807319
11	X Gu, R Guo, F Luo, J Sun, T Wu, A discrete uniformization theorem for polyhedral surfaces, II, J. Differential Geom. 109 (2018) 431 MR3825607
12	Z X He, O Schramm, Fixed points, Koebe uniformization and circle packings, Ann. of Math. 137 (1993) 369 MR1207210
13	Z X He, O Schramm, On the convergence of circle packings to the Riemann map, Invent. Math. 125 (1996) 285 MR1395721
14	S Hersonsky, Discrete harmonic maps and convergence to conformal maps, I : Combinatorial harmonic coordinates, Comment. Math. Helv. 90 (2015) 325 MR3351747
15	S Hersonsky, Electrical networks and Stephenson’s conjecture, preprint (2015) arXiv:1506.07549
16	S Y Lan, D Q Dai, The C^∞–convergence of SG circle patterns to the Riemann mapping, J. Math. Anal. Appl. 332 (2007) 1351 MR2324342
17	O Lehto, K I Virtanen, Quasiconformal mappings in the plane, 126, Springer (1973) MR0344463
18	F Luo, Combinatorial Yamabe flow on surfaces, Commun. Contemp. Math. 6 (2004) 765 MR2100762
19	F Luo, T Wu, Koebe conjecture and the Weyl problem for convex surfaces in hyperbolic 3–space, preprint (2019) arXiv:1910.08001
20	D Matthes, Convergence in discrete Cauchy problems and applications to circle patterns, Conform. Geom. Dyn. 9 (2005) 1 MR2133803
21	B Palka, Fréchet distance and the uniform convergence of quasiconformal mappings, Duke Math. J. 39 (1972) 289 MR304653
22	A V Pogorelov, Extrinsic geometry of convex surfaces, 35, Amer. Math. Soc. (1973) MR0346714
23	C Pommerenke, Univalent functions, 25, Vandenhoeck & Ruprecht (1975) 376 MR0507768
24	I Rivin, Intrinsic geometry of convex ideal polyhedra in hyperbolic 3–space, from: "Analysis, algebra, and computers in mathematical research" (editors M Gyllenberg, L E Persson), Lect. Notes Pure Appl. Math. 156, Dekker (1994) 275 MR1280952
25	B Rodin, D Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987) 349 MR906396
26	J M Schlenker, Réalisations de surfaces hyperboliques complètes dans H³, Ann. Inst. Fourier (Grenoble) 48 (1998) 837 MR1644101
27	W Thurston, Geometry and topology of three-manifolds, lecture notes (1980)
28	T Wu, X Gu, J Sun, Rigidity of infinite hexagonal triangulation of the plane, Trans. Amer. Math. Soc. 367 (2015) 6539 MR3356946