On strong convergence of hyperbolic 3-cone-manifolds whose singular sets have uniformly thick tubular neighborhoods (Q1612785)

For a compact hyperbolic 3-cone-manifold \(C\) (that is a 3-manifold with a hyperbolic metric with cone-type singularities along a set of simple closed geodesics), it is shown that any sequence \(C_i\), \(i \in \mathbb N\), of deformations of \(C\) has a subsequence converging strongly to a hyperbolic 3-cone-manifold homeomorphic to \(C\), under the hypotheses that all cone angles around the singular sets of the \(C_i\) are less than \(2\pi\) and that these singular sets have uniformly thick tubular neighbourhoods. In particular, there is a continuous angle decreasing deformation of \(C\) to the complete hyperbolic structure on the complement of the singular set of \(C\) if one can rule out the case that the singular set intersects itself during the deformation. This generalizes some results in a paper of \textit{S. Kojima} [J. Differ. Geom. 49, No. 3, 469-516 (1998; Zbl 0990.57004)] obtained for cone angles less than \(\pi\).