Infinite Ideal Polyhedra in Hyperbolic 3-Space: Existence and Rigidity

TL;DR

This paper characterizes infinite ideal polyhedra in hyperbolic 3-space using ideal circle patterns, establishing their existence, rigidity, and a uniformization theorem, thus solving key open problems in the field.

Contribution

It introduces a new approach to infinite hyperbolic polyhedra via ideal circle patterns, including existence, rigidity, and a uniformization theorem, with novel insights into angle-dependent type theory.

Findings

Established existence and rigidity of embedded ideal circle patterns on the plane.

Proved a uniformization theorem for infinite ideal circle patterns.

Demonstrated that the type theory depends on both cellular structure and intersection angles.

Abstract

In the seminal work [27], Rivin obtained a complete characterization of finite ideal polyhedra in hyperbolic 3-space by the exterior dihedral angles. Since then,the characterization of infinite hyperbolic polyhedra has become an extremely challenging open problem. By studying ideal circle patterns (ICPs), we characterize the infinite ideal polyhedra (IIP) and resolve this problem. Specifically, we establish the existence and rigidity of embedded ICPs on the plane. We further prove the uniformization theorem for the embedded ICPs, which solves the type problem of infinite ICPs. This is an analog of the uniformization theorem obtained by He and Schramm in [22, 23]. Moreover, we demonstrate that, unlike He-Schramm's work, the type theory for infinite ICPs depends not only on the structure of the cellular decomposition but also on the selection of intersection angles. In fact, we construct Example 4.13 to show the difference. Consequently, we obtain the existence and rigidity of IIP with prescribed exterior angles. Due to the example, our results on the type problem of infinite ICPs and the existence of IIP are sharp. For ICPs with arbitrary angles, our example also demonstrates that the VEL-parabolicity and ICP-parabolicity are not equivalent (while in He and Schramm's settings, VEL-parabolicity and CP-parabolicity are equivalent), indicating that our setting is extremely distinct from He and Schramm's. To prove our results, we develop a uniform Ring Lemma via the technique of pointed Gromov-Hausdorff convergence for ICPs.