## Journal of Mathematics of Kyoto University

- J. Math. Kyoto Univ.
- Volume 41, Number 2 (2001), 421-428.

### 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 $C$.

#### Article information

**Source**

J. Math. Kyoto Univ., Volume 41, Number 2 (2001), 421-428.

**Dates**

First available in Project Euclid: 17 August 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.kjm/1250517641

**Digital Object Identifier**

doi:10.1215/kjm/1250517641

**Mathematical Reviews number (MathSciNet)**

MR1852992

**Zentralblatt MATH identifier**

1002.57040

**Subjects**

Primary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

Secondary: 30F40: Kleinian groups [See also 20H10] 57M50: Geometric structures on low-dimensional manifolds 57N10: Topology of general 3-manifolds [See also 57Mxx]

#### Citation

Fujii, Michihiko. On strong convergence of hyperbolic 3-cone-manifolds whose singular sets have uniformly thick tubular neighborhoods. J. Math. Kyoto Univ. 41 (2001), no. 2, 421--428. doi:10.1215/kjm/1250517641. https://projecteuclid.org/euclid.kjm/1250517641