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The
convex core of quasifuchsian manifolds with particles
Cyril Lecuire and Jean-Marc Schlenker |
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Geometry & Topology 18 (2014) 2309–2373
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Abstract |
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We consider quasifuchsian manifolds with “particles”, ie cone singularities of fixed angle less than going from one connected component of the boundary at infinity to the other. Each connected component of the boundary at infinity is then endowed with a conformal structure marked by the endpoints of the particles. We prove that this defines a homeomorphism between the space of quasifuchsian metrics with particles (of fixed angle) and the product of two copies of the Teichmüller space of a surface with marked points. This extends the Bers double uniformization theorem to quasifuchsian manifolds with “particles”. Quasifuchsian manifolds with particles also have a convex core. Its boundary has a hyperbolic induced metric, with cone singularities at the intersection with the particles, and is pleated along a measured geodesic lamination. We prove that any two hyperbolic metrics with cone singularities (of prescribed angle) can be obtained, and also that any two measured bending laminations, satisfying some obviously necessary conditions, can be obtained, as in Bonahon and Otal [Ann. of Math. 160 (2004) 1013–1055] in the nonsingular case. |
Keywords
quasifuchsian groups, cone singularities, conformal
structures, bending laminations
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Mathematical Subject Classification 2010
Primary: 20H10, 57M50
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References |
Publication
Received: 18 February 2013
Accepted: 5 October 2013
Published: 2 October 2014
Proposed: Benson Farb
Seconded: David Gabai, Jean-Pierre Otal
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Authors |
| Cyril Lecuire | |
| Institut de Mathématiques de
Toulouse UMR CNRS 5219 Université Toulouse III 31062 Toulouse Cedex 9 France |
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| Jean-Marc Schlenker | |
| Mathematics Research Unit, BLG University of Luxembourg, Campus Kirchberg 6 rue Richard Coudenhove-Kalergi L-1359 Luxembourg Luxembourg |
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