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Ideal Hyperbolic Polyhedra and Discrete Uniformization

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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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Acknowledgements

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

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  1. Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Boris Springborn

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  1. Boris Springborn
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Correspondence to Boris Springborn.

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Springborn, B. Ideal Hyperbolic Polyhedra and Discrete Uniformization. Discrete Comput Geom 64, 63–108 (2020). https://doi.org/10.1007/s00454-019-00132-8

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