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

Abstract

For 3-dimensional hyperbolic cone structures with cone angles θ \theta , local rigidity is known for 0 ≤ θ ≤ 2 π 0 \leq \theta \leq 2\pi , but global rigidity is known only for 0 ≤ θ ≤ π 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 π \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 π 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.