Ideal hyperbolic polyhedra and discrete uniformization

TL;DR

The paper develops a constructive variational framework to address Rivin’s realization theorem for convex ideal hyperbolic polyhedra with prescribed intrinsic metrics and its connection to discrete uniformization on the sphere. By formulating a convex objective on decorated Teichmüller spaces and using Epstein–Penner hulls together with ideal Delaunay decompositions, it obtains existence and uniqueness results via minimality of a carefully designed functional. It also grounds the discrete conformal map viewpoint, showing the realization problem is equivalent to a discrete uniformization problem for spheres, with higher-genus generalizations and cone-angle prescriptions following from the same variational philosophy. The approach provides a practical, constructive pathway to compute uniformizing polyhedra by solving convex optimization problems, and it clarifies the relationships between polyhedral realizations, Delaunay theory, and discrete conformal geometry. Overall, the work unifies realization theory and discrete uniformization in a robust, computationally amenable framework that extends to higher genus cases and prescribed cone angles.

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.

Figures (14)

• Figure 1: Ideal polyhedron (bounded by the transparent hyperbolic planes) decorated with horospheres (white) at ideal vertices
• Figure 2: Signed distance $\lambda$ of disjoint and intersecting horocycles
• Figure 3: Ideal tetrahedron decorated with horospheres at the vertices. The horosphere at $v_{\infty}$ is the horizontal plane at height $1$. The horospheres at $v_{1}$, $v_{2}$, $v_{3}$ touch the horosphere at $v_{\infty}$. The horosphere at $v_{\infty}$ intersects the tetrahedron in a euclidean triangle with sides $\ell_{i}=e^{\frac{1}{2}\lambda_{i}}$, where $\lambda_{i}$ are the signed distances between horospheres (see Figure \ref{['fig:lambdaell']}).
• Figure 4: Signed distance $\lambda$ and horocyclic arc length $\ell$ in a configuration of two horocycles that are tangent to a third horocycle
• Figure 5: Decorated ideal triangle in the Poincaré disk model (left) and in the half-plane model (right)

Theorems & Definitions (83)

• Theorem 1.1: Rivin rivin94:_intrin
• Remark 2.2: shear coordinates
• Remark 2.4: Hilbert--Einstein functional
• Remark 3.1: notation warning
• Proposition 3.2: parametrizing a fiber of $\widetilde{\mathcal{T}}_{g,n}$
• Proposition 3.3: horocyclic arcs in a decorated triangle
• Proposition 3.4: horocycle length at a cusp
• Proposition 3.5: Penner coordinates and shear coordinates
• Proposition 3.6: Ptolemy relation
• proof