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Ideal Hyperbolic Polyhedra and Discrete Uniformization


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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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).

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

Mathematics Subject Classification

  • 57M50
  • 52B10
  • 52C26