Ideal hyperbolic polyhedra and discrete uniformization

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.