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Ideal Hyperbolic Polyhedra and Discrete Uniformization

Abstract

We provide a constructive, variational proof of Rivin’s realization theorem for ideal hyperbolic polyhedra with prescribed intrinsic metric, which is equivalent to a discrete uniformization theorem for spheres. The same variational method is also used to prove a discrete uniformization theorem of Gu et al. and a corresponding polyhedral realization result of Fillastre. The variational principles involve twice continuously differentiable functions on the decorated Teichmüller spaces \(\widetilde{\mathscr {T}}_{g,n}\) of punctured surfaces, which are analytic in each Penner cell, convex on each fiber over \(\mathscr {T}_{g,n}\), and invariant under the action of the mapping class group.

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References

  1. 1.

    Akiyoshi, H.: Finiteness of polyhedral decompositions of cusped hyperbolic manifolds obtained by the Epstein–Penner’s method. Proc. Am. Math. Soc. 129(8), 2431–2439 (2001)

  2. 2.

    Alexandrov, A.D.: Convex Polyhedra. Springer Monographs in Mathematics. Springer, Berlin (2005)

  3. 3.

    Aurenhammer, F., Klein, R., Lee, D.-T.: Voronoi Diagrams and Delaunay Triangulations. World Scientific, Hackensack (2013)

  4. 4.

    Bao, X., Bonahon, F.: Hyperideal polyhedra in hyperbolic 3-space. Bull. Soc. Math. France 130(3), 457–491 (2002)

  5. 5.

    Bobenko, A.I., Izmestiev, I.: Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes. Ann. Inst. Fourier (Grenoble) 58(2), 447–505 (2008)

  6. 6.

    Bobenko, A.I., Dimitrov, N., Sechelmann, S.: Discrete uniformization of polyhedral surfaces with non-positive curvature and branched covers over the sphere via hyper-ideal circle patterns. Discrete Comput. Geom. 57(2), 431–469 (2017)

  7. 7.

    Bobenko, A.I., Pinkall, U., Springborn, B.A.: Discrete conformal maps and ideal hyperbolic polyhedra. Geom. Topol. 19(4), 2155–2215 (2015)

  8. 8.

    Bobenko, A.I., Sechelmann, S., Springborn, B.: Discrete conformal maps: boundary value problems, circle domains, Fuchsian and Schottky uniformization. In: Bobenko, A.I. (ed.) Advances in Discrete Differential Geometry, pp. 1–56. Springer, Berlin (2016)

  9. 9.

    Bobenko, A.I., Springborn, B.A.: A discrete Laplace–Beltrami operator for simplicial surfaces. Discrete Comput. Geom. 38(4), 740–756 (2007)

  10. 10.

    Bowers, J.C., Bowers, P.L., Pratt, K.: Rigidity of circle polyhedra in the 2-sphere and of hyperideal polyhedra in hyperbolic 3-space. Trans. Am. Math. Soc. 371(6), 4215–4249 (2019)

  11. 11.

    Duffin, R.J.: Distributed and lumped networks. J. Math. Mech. 8, 793–826 (1959)

  12. 12.

    Edelsbrunner, H.: Geometry and Topology for Mesh Generation. Cambridge Monographs on Applied and Computational Mathematics, vol. 7. Cambridge University Press, Cambridge (2001)

  13. 13.

    Epstein, D.B.A., Penner, R.C.: Euclidean decompositions of noncompact hyperbolic manifolds. J. Differ. Geom. 27(1), 67–80 (1988)

  14. 14.

    Fillastre, F.: Polyhedral hyperbolic metrics on surfaces. Geom. Dedicata 134, 177–196 (2008). Erratum 138, 193–194 (2009)

  15. 15.

    Fillastre, F., Izmestiev, I.: Hyperbolic cusps with convex polyhedral boundary. Geom. Topol. 13(1), 457–492 (2009)

  16. 16.

    Fortune, S.: Numerical stability of algorithms for 2D Delaunay triangulations. Int. J. Comput. Geom. Appl. 5(1–2), 193–213 (1995)

  17. 17.

    Gu, X., Guo, R., Luo, F., Sun, J., Wu, T.: A discrete uniformization theorem for polyhedral surfaces II. J. Differ. Geom. 109(3), 431–466 (2018)

  18. 18.

    Gu, X.D., Luo, F., Sun, J., Wu, T.: A discrete uniformization theorem for polyhedral surfaces. J. Differ. Geom. 109(2), 223–256 (2018)

  19. 19.

    Indermitte, C., Liebling, T.M., Troyanov, M., Clémençon, H.: Voronoi diagrams on piecewise flat surfaces and an application to biological growth. Theoret. Comput. Sci. 263(1–2), 263–274 (2001)

  20. 20.

    Izmestiev, I.: A variational proof of Alexandrov’s convex cap theorem. Discrete Comput. Geom. 40(4), 561–585 (2008)

  21. 21.

    Joswig, M., Löwe, R., Springborn, B.: Secondary fans and secondary polyhedra of punctured Riemann surfaces. Exp. Math. (2019). https://doi.org/10.1080/10586458.2018.1477078

  22. 22.

    Luo, F.: Combinatorial Yamabe flow on surfaces. Commun. Contemp. Math. 6(5), 765–780 (2004)

  23. 23.

    Masur, H., Smillie, J.: Hausdorff dimension of sets of nonergodic measured foliations. Ann. Math. 134(3), 455–543 (1991)

  24. 24.

    Milnor, J.: Hyperbolic geometry: the first 150 years. Bull. Am. Math. Soc. (N.S.) 6(1), 9–24 (1982)

  25. 25.

    Moroianu, S., Schlenker, J.-M.: Quasi-Fuchsian manifolds with particles. J. Differ. Geom. 83(1), 75–129 (2009)

  26. 26.

    Penner, R.C.: The decorated Teichmüller space of punctured surfaces. Commun. Math. Phys. 113(2), 299–339 (1987)

  27. 27.

    Penner, R.C.: Decorated Teichmüller Theory. QGM Master Class Series. European Mathematical Society (EMS), Zürich (2012)

  28. 28.

    Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Exp. Math. 2(1), 15–36 (1993)

  29. 29.

    Prosanov, R.: Ideal polyhedral surfaces in Fuchsian manifolds. arXiv:1804.05893 (2018)

  30. 30.

    Rippa, S.: Minimal roughness property of the Delaunay triangulation. Comput. Aided Geom. Des. 7(6), 489–497 (1990)

  31. 31.

    Rivin, I.: Intrinsic geometry of convex ideal polyhedra in hyperbolic \(3\)-space. In: Gyllenberg, M., Persson, L.-E. (eds.) Analysis, Algebra, and Computers in Mathematical Research. Lecture Notes in Pure and Appl. Math., vol. 156, pp. 275–291. Dekker, New York (1994)

  32. 32.

    Sakuma, M., Weeks, J.R.: The generalized tilt formula. Geom. Dedicata 55(2), 115–123 (1995)

  33. 33.

    Schlenker, J.-M.: Hyperbolic manifolds with polyhedral boundary. arXiv:math/0111136 (2001)

  34. 34.

    Schlenker, J.-M.: A rigidity criterion for non-convex polyhedra. Discrete Comput. Geom. 33(2), 207–221 (2005)

  35. 35.

    Springborn, B.A.: A variational principle for weighted Delaunay triangulations and hyperideal polyhedra. J. Differ. Geom. 78(2), 333–367 (2008)

  36. 36.

    Thurston, W.P.: Shapes of polyhedra and triangulations of the sphere. In: Rivin, I., Rourke, C., Series, C. (eds.) The Epstein Birthday Schrift. Geometry & Topology Monographs, vol. 1, pp. 511–549. Geometry & Topology Publications, Coventry (1998)

  37. 37.

    Tillmann, S., Wong, S.: An algorithm for the Euclidean cell decomposition of a cusped strictly convex projective surface. J. Comput. Geom. 7(1), 237–255 (2016)

  38. 38.

    Weeks, J.R.: Convex hulls and isometries of cusped hyperbolic \(3\)-manifolds. Topology Appl. 52(2), 127–149 (1993)

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Acknowledgements

This research was supported by DFG SFB/Transregio 109 “Discretization in Geometry and Dynamics”.

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Correspondence to Boris Springborn.

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Springborn, B. Ideal Hyperbolic Polyhedra and Discrete Uniformization. Discrete Comput Geom (2019). https://doi.org/10.1007/s00454-019-00132-8

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Keywords

  • Decorated Teichmüller space
  • Penner coordinates
  • Horocycle
  • Discrete conformal equivalence
  • Triangulated surface

Mathematics Subject Classification

  • 57M50
  • 52B10
  • 52C26