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 $$\widetilde{\mathscr {T}}_{g,n}$$ T ~ g , n of punctured surfaces, which are analytic in each Penner cell, convex on each fiber over $$\mathscr {T}_{g,n}$$ T g , n , and invariant under the action of the mapping class group.

中文翻译：

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理想双曲多面体和离散均匀化

我们提供了具有规定内在度量的理想双曲多面体的 Rivin 实现定理的建设性变分证明，这等效于球体的离散均匀化定理。同样的变分方法也用于证明 Gu 等人的离散均匀化定理。以及相应的Fillastre多面体实现结果。变分原理涉及装饰 Teichmüller 空间 $$\widetilde{\mathscr {T}}_{g,n}$$ T ~ g , n 个穿孔表面上的两次连续可微函数，它们在每个 Penner 单元中是解析的，凸面在 $$\mathscr {T}_{g,n}$$ T g , n 上的每根光纤上，在映射类组的作用下不变量。