[Paper Review] Intrinsic geometry of convex ideal polyhedra in hyperbolic 3-space

Igor Rivin|ArXiv.org|May 23, 2000

Geometric and Algebraic Topology3 references21 citations

TL;DR

This paper establishes a complete intrinsic characterization of convex ideal polyhedra in hyperbolic 3-space, proving that such polyhedra are uniquely determined by their intrinsic metric up to congruence. It uses the invariance of domain principle and shear parameters in triangulations to show that every complete finite-volume hyperbolic surface homeomorphic to the N-punctured sphere admits a unique convex ideal polyhedral embedding in H³ with vertices at infinity.

ABSTRACT

The main result is that every complete finite area hyperbolic metric on a sphere with punctures can be uniquely realized as the induced metric on the surface of a convex ideal polyhedron in hyperbolic 3-space. A number of other observations are included.

Motivation & Objective

• To provide a complete intrinsic characterization of convex ideal polyhedra in hyperbolic 3-space with all vertices at infinity.
• To establish that such polyhedra are uniquely realizable up to congruence from their intrinsic metric.
• To develop a framework linking hyperbolic triangulations and polyhedral structures via shear parameters.
• To extend the uniqueness result to generalized polyhedra with finite, ideal, and hyperinfinite vertices.
• To lay the foundation for constructive methods of embedding N-punctured hyperbolic surfaces as ideal polyhedra.

Proposed method

• Uses the invariance of domain principle to prove surjectivity of the map from polyhedral embeddings to hyperbolic structures on the N-punctured sphere.
• Parametrizes the space of ideal polyhedra via vertex positions on the Riemann sphere, fixing three vertices to eliminate isometry group action.
• Introduces shear parameters between adjacent ideal triangles in a geodesic triangulation to encode geometric data.
• Defines links of vertices (spherical, hyperbolic, or Euclidean) to capture local geometry at finite, hyperinfinite, and ideal vertices.
• Applies Theorem 4.4, which states that generalized polyhedra are determined up to congruence by vertex types and link edge lengths.
• Establishes continuity and closedness of the metric map from polyhedral embeddings to Teichmüller space, enabling topological uniqueness.

Experimental results

Research questions

• RQ1 Can every complete finite-volume hyperbolic surface homeomorphic to the N-times punctured sphere be isometrically embedded as a convex ideal polyhedron in H³?
• RQ2 Is such an embedding unique up to congruence, given only the intrinsic metric of the surface?
• RQ3 How can the geometric data of an ideal polyhedron be encoded intrinsically using triangulations and shear parameters?
• RQ4 To what extent do the links of vertices (spherical, hyperbolic, or Euclidean) determine the global structure of a generalized hyperbolic polyhedron?
• RQ5 Can the intrinsic metric of an ideal polyhedron be used to reconstruct its full geometric realization in H³?

Key findings

• Every complete finite-volume hyperbolic surface homeomorphic to the N-punctured sphere admits a unique convex ideal polyhedral embedding in H³ with all vertices on the sphere at infinity.
• The intrinsic metric of a convex ideal polyhedron uniquely determines its geometric realization up to congruence in H³.
• Shear parameters between adjacent ideal triangles in a triangulation encode the dihedral geometry of the polyhedron, with the logarithm of side length ratios in vertex links equaling the shear.
• Generalized polyhedra in H³ are determined up to congruence by the types of their vertices and the edge lengths in their vertex links.
• The space of convex ideal polyhedra with N vertices is a 2N−6 dimensional manifold, matching the dimension of the Teichmüller space of the N-punctured sphere.
• The map from polyhedral embeddings to hyperbolic structures on the N-punctured sphere is continuous, open, and closed, ensuring surjectivity and uniqueness via the invariance of domain principle.