Ideal hyperbolic polyhedra and discrete uniformization
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2017
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| _version_ | 1866912176824909824 |
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| author | Springborn, Boris |
| author_facet | Springborn, Boris |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1707_06848 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Ideal hyperbolic polyhedra and discrete uniformization Springborn, Boris Metric Geometry Geometric Topology 57M50, 52B10, 52C26 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. |
| title | Ideal hyperbolic polyhedra and discrete uniformization |
| topic | Metric Geometry Geometric Topology 57M50, 52B10, 52C26 |
| url | https://arxiv.org/abs/1707.06848 |