Deformation of three-dimensional hyperbolic cone structures: the noncollapsing case

Alexandre Paiva Barreto

Vol. 268 (2014), No. 1, 1–21

DOI: 10.2140/pjm.2014.268.1

Abstract

This work is devoted to the study of deformations of hyperbolic cone structures under the assumption that the length of the singularity remains uniformly bounded over the deformation. Let $\left(M_i,p_i\right)$ be a sequence of pointed hyperbolic cone manifolds with cone angles of at most $2\pi$ and topological type $\left(M,\Sigma \right)$, where $M$ is a closed, orientable and irreducible $3$-manifold and $\Sigma$ an embedded link in $M$. Assuming that the length of the singularity remains uniformly bounded, we prove that either the sequence $M_i$ collapses and $M$ is Seifert fibered or a $Sol$ manifold, or the sequence $M_i$ does not collapse and, in this case, a subsequence of $\left(M_i,p_i\right)$ converges to a complete three dimensional Alexandrov space endowed with a hyperbolic metric of finite volume on the complement of a finite union of quasigeodesics. We apply this result to a question proposed by Thurston and to provide universal constants for hyperbolic cone structures when $\Sigma$ is a small link in $M$.

Dedicated to my wife Cynthia

Keywords

hyperbolic $3$-manifolds, cone manifolds, Alexandrov spaces

Mathematical Subject Classification 2010

Primary: 57M50

Secondary: 57N16, 53C23

Milestones

Received: 19 January 2012

Revised: 23 October 2013

Accepted: 26 October 2013

Published: 21 May 2014