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

Abstract

For deformation of 3-dimensional hyperbolic cone structures about cone angles $\theta$, the local rigidity is known for $0 \leq \theta < 2\pi$, but the 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 do not degenerate in deformation with decreasing cone angles at most $\pi$. In this paper, we give an example of 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 polyhedra. For this construction, we explicitly describe the isometry types on such a hyperbolic polyhedron. Cone loci intersect in our example of degeneration. In order to avoid such degeneration, we generalize a cone metric to a holed cone metric.Comment: 11pages, 4 figures. v2: Section 4 adde