Hyperbolic polyhedra and discrete uniformization
| Autor: | Springborn, Boris |
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| Rok vydání: | 2017 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 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 |
| Databáze: | arXiv |
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