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

Huabin Ge, Hao Yu, Puchun Zhou

TL;DR

This work extends Rivin's finite-characterization of ideal hyperbolic polyhedra to the infinite setting by recasting the problem in terms of embedded ideal circle patterns (ICPs) on the plane. It develops a discrete uniformization framework using vertex extremal length (VEL), Ring Lemmas for ICPs, and pointed Gromov–Hausdorff convergence, yielding existence, a uniformization theorem, and rigidity results. Key contributions include existence of embedded ICPs, a precise type criterion distinguishing ICP-parabolic/hyperbolic (and PIIP/HIIP) under a uniform angle bound, and rigidity of IIP at fixed dihedral angles, with sharpness shown through constructed counterexamples when angle bounds are weakened. These results resolve Rivin's questions about infinite ideal polyhedra and illuminate important differences from He–Schramm's setting when intersection angles are generalized, thereby providing a complete and sharp discrete-analytic framework for infinite hyperbolic polyhedra.

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.

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

TL;DR

This work extends Rivin's finite-characterization of ideal hyperbolic polyhedra to the infinite setting by recasting the problem in terms of embedded ideal circle patterns (ICPs) on the plane. It develops a discrete uniformization framework using vertex extremal length (VEL), Ring Lemmas for ICPs, and pointed Gromov–Hausdorff convergence, yielding existence, a uniformization theorem, and rigidity results. Key contributions include existence of embedded ICPs, a precise type criterion distinguishing ICP-parabolic/hyperbolic (and PIIP/HIIP) under a uniform angle bound, and rigidity of IIP at fixed dihedral angles, with sharpness shown through constructed counterexamples when angle bounds are weakened. These results resolve Rivin's questions about infinite ideal polyhedra and illuminate important differences from He–Schramm's setting when intersection angles are generalized, thereby providing a complete and sharp discrete-analytic framework for infinite hyperbolic polyhedra.

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.

Paper Structure

This paper contains 20 sections, 44 theorems, 133 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Gromov-Hausdorff convergence on graphs
  4. Ideal circle patterns and ideal polyhedra
  5. Limit structures of ICPs
  6. The limits of embedded ICPs
  7. Ring lemmas for ICPs
  8. The existence of infinite ICPs
  9. Vertex extremal length
  10. Geometric behaviors of embedded ICPs
  11. Proof of the uniformization theorem
  12. CP-hyperbolic v.s. ICP-hyperbolic
  13. The rigidity of ICPs
  14. Networks and harmonic function
  15. Variational principles and maximum principles for ICPs
  16. ...and 5 more sections

Key Result

Theorem 1.0

(Rivin) Let $\mathcal{D}=(V,E,F)$ be a finite cellular decomposition of the sphere $\mathbb{S}^2$ and let $\Theta\in(0,\pi)^E$, then there exists an ideal polyhedron $P$ which is combinatorially equivalent to the Poincaré dual of $\mathcal{D}$ with dihedral angle $\Theta(e^*)=\Theta(e)$ if and only Moreover, the ideal hyperbolic polyhedron is unique up to isometry.

Figures (11)

  • Figure 1: A portion of an ideal circle pattern in the plane.
  • Figure 2: An Euclidean quadrilateral $\tilde{Q}_e$.
  • Figure 3: A subgraph $G_0$ of a cellular decomposition and its reduced graph.
  • Figure 4: Half of the quadrilaterals $\{\tilde{Q}_{(\phi^{-1}_i(e_1))},\cdots,\tilde{Q}_{(\phi^{-1}_i(e_N))}\}$ in the embedded ICP $\mathcal{P}_i$.
  • Figure 5: A maple $\mathcal{M}(v)$.
  • ...and 6 more figures

Theorems & Definitions (86)

  • Theorem 1.0
  • Theorem 1.1: Existence of ICPs
  • Theorem 1.2: Existence of IIP
  • Definition 1.3
  • Theorem 1.4: Uniformization of ICPs and IIP
  • Remark 1.5
  • Theorem 1.6: Rigidity of parabolic ICPs
  • Theorem 1.7: Rigidity of hyperbolic ICPs
  • Theorem 1.8: Rigidity of IIP
  • Definition 2.1: Pointed Gromov–Hausdorff convergence
  • ...and 76 more