Geometry III: Discrete conformal maps (Winter 2022/23)

[Bobenko/Sechelmann/Springborn]	[Gillespie/Springborn/Crane]

Lectures

Boris Springborn

This is a course of the Berlin Mathematical School held in English unless all attending BMS students prefer German.

Contents

The course will be on the following topics and their connections:

• Nonlinear theory of discrete conformal maps:
  • circle packings and circle patterns
  • discrete conformal equivalence of triangle meshes
• Realization of hyperpolic polyhedra with prescribed dihedral angles, or prescribed intrinsic metric.

Movies

• conform. A short film about conformal maps. [youtube, vimeo]
• conform. Ein kurzer Film über konforme Abbildungen. (deutsch) [vimeo]
• Koebe polyhedra and minimal surfaces. [youtube]

Literature

• Circle packings, circle patterns, ideal hyperbolic polyhedra with prescribed dihedral angles, discrete conformal maps via circumcircle intersection angles
• A. I. Bobenko, B. Springborn. Variational principles for circle patterns and Koebe's theorem. Trans. Amer. Math. Soc. 356 (2004), 659-689.
  
• Y. Colin de Verdière. Un principe variationnel pour les empilements de cercles. Invent. Math. 104 (1991), no. 1, 655-669.
• F. Guéritaud. On an elementary proof of Rivin's characterization of convex ideal hyperbolic polyhedra by their dihedral angles. Geometriae Dedicata 108 (2004), no. 1, 111-124.
• I. Rivin. A characterization of ideal polyhedra in hyperbolic 3-space, Ann. of Math. (2), 143 (1996), no. 1, 51-70.
• I. Rivin. Euclidean structures on simplicial surfaces and hyperbolic volume. Ann. of Math. (2), 139 (1994), no. 1, 553-580.
• I. Rivin. Combinatorial optimization in geometry. Adv. in Appl. Math. 31 (2003), no. 1, 242-271. arXiv
• B. Springborn. A unique representation of polyhedral types. Centering via Möbius transformations. Math. Z., 249(3):513--517, 2005.
• B. Springborn. Variational principles for circle patterns. Doktorarbeit, TU Berlin, 2003.
• W. P. Thurston. The Geometry and Topology of Three-Manifolds, electronic edition of the 1980 lecture notes distributed by Princeton University.
• Discrete conformal maps via length scaling, ideal hyperbolic polyhedra with prescribed intrinsic metric
• A. Bobenko, U. Pinkall, B. Springborn. Discrete conformal maps and ideal hyperbolic polyhedra. Geom. Topol. 19-4 (2015) 2155-2215. arXiv:1005.2698.
• A. I. Bobenko, S. Sechelmann, and B. Springborn. Discrete conformal maps: boundary value problems, circle domains, Fuchsian and Schottky uniformization. In A. I. Bobenko, editor, Advances in Discrete Differential Geometry, pages 1--56. Springer, Berlin, 2016.
• F. Luo. Combinatorial Yamabe flow on surfaces.Commun. Contemp. Math. 6 (2004), no. 5, 765–780.
• M. Gillespie, B. Springborn, and K. Crane. Discrete conformal equivalence of polyhedral surfaces. ACM Trans. Graph., 40(4), 2021.
• X. D. Gu, F. Luo, J. Sun, T. Wu. A discrete uniformization theorem for polyhedral surfaces. J. Differential Geom. 109 (2018), no. 2, 223–256.
• B. Springborn. Ideal Hyperbolic Polyhedra and Discrete Uniformization. Discrete Comput. Geom. (2019)
• General / related / background
• D. V. Alekseevskij, E. B. Vinberg, A. S. Solodovnikov. Geometry of spaces of constant curvature. In: E. B. Vinberg (editor). Geometry II. Encyclopedia of Mathematical Sciences 29. Springer, Berlin, 1993. Pages 1-138.
• J. Böhm, E. Hertel. Polyedergeometrie in n-dimensionalen Räumen konstanter Krümmung. Mathematische Monographien, Bd. 14. VEB Deutscher Verlag der Wissenschaften, Berlin, 1980. Lizenzausgabe bei Birkhäuser 1981.
• H. Kneser. Der Simplexinhalt in der nichteuklidischen Geometrie. Deutsche Mathematik 1 (1936), 337-334. (good paper in nasty journal)
• L. Lewin. Polylogarithms and associated functions. North-Holland Publishing Co., New York-Amsterdam, 1981.
• J. Milnor. Hyperbolic geometry: The first 150 years. Bull. Amer. Math. Soc. (N.S.) 6, (1982), no. 1, 9-24.
• J. Milnor. The Schläfli differential equalitiy. In: Collected Papers, volume 1, pages 281-295. Publish or Perish Inc., Houston, TX, 1994.
• M. Passare. How to compute Σ1/n² by solving integrals. Amer. Math. Monthly 115 (2008), no. 8, 745-752.
• D. Zagier. The dilogarithm function. In: Frontiers in number theory, physics, and geometry. II, 3-65, Springer, Berlin, 2007.

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