Ideal Hyperbolic Polyhedra and Discrete Uniformization

  • Boris SpringbornEmail author


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.


Decorated Teichmüller space Penner coordinates Horocycle Discrete conformal equivalence Triangulated surface 

Mathematics Subject Classification

57M50 52B10 52C26 



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


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

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