On strong convergence of hyperbolic 3-cone-manifolds whose singular sets have uniformly thick tubular neighborhoods

Abstract

Let $C$ be a compact orientable hyperbolic 3-cone-manifold with cone-type singularity along simple closed geodesics $\Sigma$. Let $\{ C_{i}\}_{i=1}^{\infty}$ be a sequence consisting of deformations of $C$ and $\Sigma _{i}$ be the singular set of $C_{i}$ so that the cone angles along $\Sigma _{i}$ all are less than $2\pi$. In this paper, we will show that, if tubular neighborhoods of the singular sets $\Sigma _{i}$ can be taken to be uniformly thick, then there is a subsequence $\{ C_{i_{k}}\}_{k=1}^{\infty}$ which converges strongly to a hyperbolic 3-cone-manifold $C_{*}$ homeomorphic to $C$.