Degeneration of 3-dimensional hyperbolic cone structures with decreasing cone angles

TL;DR

This paper demonstrates an example of degeneration in 3D hyperbolic cone structures with decreasing cone angles below 2π, challenging previous assumptions about their rigidity and stability.

Contribution

It provides the first explicit example of degeneration for cone angles less than 2π, constructed on an alternating link in a thickened torus with detailed polyhedral isometry descriptions.

Findings

Degeneration occurs for cone angles less than 2π.

Explicit construction on an alternating link in a thickened torus.

Detailed description of hyperbolic polyhedron isometries.

Abstract

For 3-dimensional hyperbolic cone structures with cone angles $\theta$, local rigidity is known for $0 \leq \theta \leq 2\pi$, but global rigidity is known only for $0 \leq \theta \leq \pi$. The proof of the global rigidity by Kojima is based on the fact that hyperbolic cone structures with cone angles at most $\pi$ do not degenerate in deformations decreasing cone angles to zero. In this paper, we give an example of a degeneration of hyperbolic cone structures with decreasing cone angles less than $2\pi$. These cone structures are constructed on a certain alternating link in the thickened torus by gluing four copies of a certain polyhedron. For this construction, we explicitly describe the isometry types on such a hyperbolic polyhedron.