The convex core of quasifuchsian manifolds with particles
type: Item
instance of: scholarly article
title: The convex core of quasifuchsian manifolds with particles
zbMATH Open document ID: 1334.57015
author: Cyril Lecuire
author: Jean-Marc Schlenker
published in: Geometry \& Topology
publication date: 2014-10-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/0909.4182
reviewed by: Mikhail Belolipetsky
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 20H10
zbMATH DE Number: 6356610
zbMATH Keywords: quasifuchsian groups
zbMATH Keywords: cone singularities
zbMATH Keywords: conformal structures
zbMATH Keywords: bending laminations
MaRDI profile type: Publication
OpenAlex ID: W3100445737
arXiv ID: 0909.4182
cites work: Riemann's mapping theorem for variable metrics
cites work: Q5524290
cites work: Notes on a paper of Mess
cites work: Manifolds of nonpositive curvature
cites work: Collisions of particles in locally AdS spacetimes. I. Local description and global examples
cites work: Collisions of particles in locally AdS spacetimes. II: Moduli of globally hyperbolic spaces
cites work: Lectures on hyperbolic geometry
cites work: Simultaneous uniformization
cites work: Q4720334
cites work: Geometrization of 3-dimensional orbifolds
cites work: Geometrization of 3-orbifolds of cyclic type. With an appendix: Limit of hyperbolicity for spherical 3-orbifolds by Michael Heusener and Joan Porti
cites work: Measured pleated laminations of hyperbolic manifolds of dimension 3
cites work: AdS manifolds with particles and earthquakes on singular surfaces
cites work: Fixed points of compositions of earthquakes
cites work: Average bending of convex pleated planes in hyperbolic three-space
cites work: From the boundary of the convex core to the conformal boundary
cites work: The classification of Kleinian surface groups. II: The Ending lamination conjecture
cites work: Asymptotics of Weil-Petersson geodesics. I: Ending laminations, recurrence, and flows
cites work: Hyperbolic cone-manifolds, short geodesics, and Schwarzian derivatives
cites work: Rigidity of geometrically finite hyperbolic cone-manifolds
cites work: Collars in Kleinian groups
cites work: Q4040246
cites work: Three-dimensional orbifolds and cone-manifolds
cites work: Collars and partitions of hyperbolic cone-surfaces
cites work: Complex Projective Structures
cites work: Q3753148
cites work: The logarithmic spiral: A counterexample to the \(K=2\) conjecture
cites work: Combinatorics of Train Tracks. (AM-125)
cites work: Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery
cites work: Canonical quantization of gravitating point particles in 2+1 dimensions
cites work: Quantization of point particles in (2 + 1)-dimensional gravity and spacetime discreteness
cites work: The Nielsen realization problem
cites work: On the renormalized volume of hyperbolic 3-manifolds
cites work: A symplectic map between hyperbolic and complex Teichmüller theory
cites work: The Weil-Petersson metric and the renormalized volume of hyperbolic 3-manifolds
cites work: A canonical metric for Möbius structures and its applications
cites work: Prescribed metrics on the boundary of hyperbolic manifolds of dimension 3
cites work: Pleating of hyperbolic 3-manifolds
cites work: The moduli space of Riemann surfaces is Kähler hyperbolic
cites work: Lorentz spacetimes of constant curvature
cites work: A Lower Bound for the Volume of Hyperbolic 3-Manifolds
cites work: The classification of punctured-torus groups
cites work: Quasi-Fuchsian manifolds with particles
cites work: Q4891081
cites work: Metrics on hyperbolic convex polyhedra
cites work: Convex polyhedra in Lorentzian space-forms
cites work: Q4810683
cites work: Hyperbolic manifolds with convex boundary
cites work: Weil-Petersson geometry for families of hyperbolic conical Riemann surfaces
cites work: Q3881471
cites work: Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography
cites work: Q3784392
cites work: Hyperbolic 2-spheres with conical singularities, accessory parameters and Kähler metrics on ℳ_{0,𝓃}
cites work: Q3764899
cites work: Prescribing Curvature on Compact Surfaces with Conical Singularities
cites work: Eine Verallgemeinerung des Schleifensatzes
DOI: 10.2140/GT.2014.18.2309
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The convex core of quasifuchsian manifolds with particles (review text)
review text: A quasifuchsian manifold is a complete hyperbolic manifold \(M\) diffeomorphic to \(S \times \mathbb{R}\), where \(S\) is a closed oriented surface of genus at least \(2\), which contains a nonempty compact convex subset. Such a manifold has a boundary at infinity formed by two copies of \(S\), which come with conformal structures induced by the hyperbolic metric on \(M\). The celebrated double uniformization theorem of \textit{L. Bers} [Bull. Am. Math. Soc. 66, 94--97 (1960; Zbl 0090.05101)] asserts these two conformal structures uniquely determine the quasifuchsian metric on \(M\), thus defining a homeomorphism between the product of two copies of the Teichmüller space of \(S\) and the space of quasifuchsian metrics. The smallest nonempty convex subset of \(M\) is called the convex core \(C(M)\). Its boundary is again the union of two copies of \(S\), each being a pleated surface in \(M\) with an induced hyperbolic metric and a measured bending lamination. The well known results of \textit{F. Bonahon} and \textit{J.-P. Otal} [Ann. Math. (2) 160, No. 3, 1013--1055 (2005; Zbl 1083.57023)] establish the relation between the quasifuchsian metric on \(M\) and the induced boundary metrics and laminations.NEWLINENEWLINEThe main goal of the present paper is an extension of these results to quasifuchsian manifolds with ``particles'', i.e. cone singularities of fixed angle less than \(\pi\) along curves joining the two boundary components. Along the way the authors also obtain some new proofs of known theorems. Among the key ingredients of the proofs are a compactness lemma in Section 3 and a local rigidity result for hyperbolic cone-manifolds of Hodgson and Kerckhoff.NEWLINENEWLINEThe paper is clearly written with many details explained very carefully. Besides proving new interesting results, it may also serve as a good reference for an introduction to the subject.