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arXiv · 1707.06848

Ideal hyperbolic polyhedra and discrete uniformization

Boris Springborn
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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 Teichm\"uller spaces $\widetilde{\mathcal{T}_{g,n}}$ of punctured surfaces, which are analytic in each Penner cell, convex on each fiber over $\mathcal{T}_{g,n}$, and invariant under the action of the mapping class group.

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Boris Springborn. 2017-07-21. Ideal hyperbolic polyhedra and discrete uniformization. https://doi.org/10.1007/s00454-019-00132-8

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