Bending parameterization of geometrically finite hyperbolic manifolds
Abstract
We show that non-Fuchsian geometrically finite hyperbolic structures on a given hyperbolizable 3-manifold M are uniquely determined, up to isotopy, by their bending laminations. Consequently, the bending map from the space of non-Fuchsian geometrically finite structures on M, endowed with the strong topology, to the space of bending laminations, endowed with Lecuire's tubular topology, is a homeomorphism. This extends the convex co-compact case established with Schlenker. The proof combines hyperbolic Dehn filling, continuity and properness of the bending map (Lecuire) and real-analyticity of its fibres (Bonahon). We also establish a criterion for contractibility of the boundary fibres of a continuous extension of a homeomorphism, extending Finney's theorem to a boundary setting. This topological result may be of independent interest.
Related
- Growth of Simple Closed Geodesics in Finite-Volume Hyperbolic Manifolds
- A Novel Bijective Angle and Volume-preservation Balanced Parameterization for n-dimensional Manifolds
- Uniformization of Julia sets of geometrically finite rational maps
- Markov maps and quasi-conformal deformations associated to geometrically finite Kleinian groups
- A space-time finite element formulation for geometrically exact shear-deformable beams
BibTeX
@misc{dular2026bending,
title = {{Bending parameterization of geometrically finite hyperbolic manifolds}},
author = {Bruno Dular},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29450},
eprint = {2609.29450},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29450},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bending parameterization of geometrically finite hyperbolic manifolds
AU - Bruno Dular
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29450
UR - https://arxiv.org/abs/2609.29450
AB - We show that non-Fuchsian geometrically finite hyperbolic structures on a given hyperbolizable 3-manifold M are uniquely determined, up to isotopy, by their bending laminations. Consequently, the bending map from the space of non-Fuchsian geometrically finite structures on M, endowed with the strong topology, to the space of bending laminations, endowed with Lecuire's tubular topology, is a homeomorphism. This extends the convex co-compact case established with Schlenker. The proof combines hyperbolic Dehn filling, continuity and properness of the bending map (Lecuire) and real-analyticity of its fibres (Bonahon). We also establish a criterion for contractibility of the boundary fibres of a continuous extension of a homeomorphism, extending Finney's theorem to a boundary setting. This topological result may be of independent interest.
ER -