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 $\widetilde{{\mathcal{T}}_{g,n}}$ 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.

41 pages, 14 figures. v2: stronger differentiability statement (C^2, was C^1), convexity holds only on fibers, error in Prop. 5.15 corrected. v3: added details to proof of Lemma 8.1, small changes in exposition. v4: final version

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• Secondary fans and secondary polyhedra of punctured Riemann surfaces
• Seamless Parametrization in Penner Coordinates
• Discrete Yamabe problem for polyhedral surfaces
• A discrete version of Liouville's theorem on conformal maps
• Convergence Analysis of the Algorithm in "Efficient and Robust Discrete Conformal Equivalence with Boundary"