[Paper Review] Examples of infinitesimally flexible 3--dimensional hyperbolic cone-manifolds

Ivan Izmestiev|ArXiv.org|Oct 15, 2009

Geometric and Algebraic Topology2 references3 citations

TL;DR

This paper constructs new examples of 3-dimensional hyperbolic cone-manifolds that are infinitesimally flexible, demonstrating that the rigidity results of Weiss and Mazzeo–Montcouquiol for cone angles less than $2\pi$ are optimal. By doubling infinitesimally flexible hyperbolic polyhedra—using Pogorelov's theorem to transfer flexibility from Euclidean to hyperbolic geometry—it constructs cone-manifolds without vertices and with all cone angles exceeding $2\pi$, showing that global rigidity fails under such conditions.

ABSTRACT

Weiss and, independently, Mazzeo and Montcouquiol recently proved that a 3--dimensional hyperbolic cone-manifold (possibly with vertices) with all cone angles less than $2π$ is infinitesimally rigid. On the other hand, Casson provided 1998 an example of an infinitesimally flexible cone-manifold with some of the cone angles larger than $2π$. In this paper several new examples of infinitesimally flexible cone-manifolds are constructed. The basic idea is that the double of an infinitesimally flexible polyhedron is an infinitesimally flexible cone-manifold. With some additional effort, we are able to construct infinitesimally flexible cone-manifolds without vertices and with all cone angles larger than $2π$.

Motivation & Objective

• To demonstrate that the infinitesimal rigidity results for hyperbolic cone-manifolds with cone angles less than $2\pi$ are optimal by constructing counterexamples with angles exceeding $2\pi$.
• To extend the understanding of flexibility in hyperbolic cone-manifolds by constructing examples without vertices and with all cone angles greater than $2\pi$.
• To show that global rigidity fails when cone angles exceed $2\pi$, using the construction of non-isometric cone-manifolds with identical cone angles.
• To utilize the double of an infinitesimally flexible polyhedron as a mechanism to generate new flexible cone-manifolds while preserving cone angles.

Proposed method

• Use Pogorelov’s theorem to transfer infinitesimal flexibility from Euclidean polyhedra to their hyperbolic counterparts in the Klein model.
• Construct compact hyperbolic polyhedra by truncating non-compact hyperideal polyhedra derived from Euclidean flexible polyhedra.
• Double an infinitesimally flexible hyperbolic polyhedron to produce a cone-manifold where opposite deformations cancel at dihedral angles, preserving cone angles.
• Glue four copies of a truncated hyperideal polyhedron together to form a cone-manifold without vertices that inherits the flexibility of the original.
• Apply branched covers to increase cone angles beyond $2\pi$ while preserving the flexibility and cone angle structure.
• Use the hyperbolic version of Pogorelov’s Lemma 4.1 to generate non-congruent but isometrically deformed polyhedral pairs, enabling construction of non-isometric cone-manifolds with identical cone angles.

Experimental results

Research questions

• RQ1 Can infinitesimally flexible hyperbolic cone-manifolds be constructed with all cone angles greater than $2\pi$?
• RQ2 Is the infinitesimal rigidity result for cone-manifolds with cone angles less than $2\pi$ optimal, or can flexibility persist beyond this threshold?
• RQ3 Can cone-manifolds without vertices be made infinitesimally flexible when all cone angles exceed $2\pi$?
• RQ4 Do non-isometric cone-manifolds with identical cone angles exist when cone angles are greater than $2\pi$?
• RQ5 Can the double of a flexible polyhedron yield a flexible cone-manifold with stable cone angles?

Key findings

• A compact infinitesimally flexible hyperbolic cone-manifold homeomorphic to a 3-ball with singular locus the skeleton of an octahedron is constructed.
• Infinitesimally flexible cone-manifolds without vertices and with all cone angles greater than $2\pi$ are explicitly constructed via gluing four copies of a truncated hyperideal polyhedron.
• The double of an infinitesimally flexible hyperbolic polyhedron yields an infinitesimally flexible cone-manifold due to cancellation of dihedral angle variations.
• Non-isometric cone-manifolds with identical cone angles exist, as shown by constructing a two-parameter family of such manifolds with overlapping cone angle maps.
• By applying branched covers, all cone angles can be made larger than $2\pi$, confirming that global rigidity fails in this regime.
• The construction confirms that the rigidity theorems of Weiss and Mazzeo–Montcouquiol are optimal, as flexibility persists when cone angles exceed $2\pi$.