Rigidity of Geometrically Finite Hyperbolic Cone-Manifolds

Bromberg, Kenneth

doi:10.1023/b:geom.0000024664.84428.e7

Cited by 36 publications

(74 citation statements)

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“…Note that there is another possible notion of quasifuchsian manifolds with cone singularities: those which are singular along closed curves, as studied in particular by Bromberg [Bro04b,Bro04a]. Although there are similarities between those two kinds cone-manifolds (in particular concerning their rigidity), the questions considered here are quite different from those usually associated to those considered for quasifuchsian conemanifolds with singularities along closed curves (drilling of geodesics, etc).…”

Section: Definition 14mentioning

confidence: 96%

The convex core of quasifuchsian manifolds with particles

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Schlenker

2014

Geom. Topol.

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We consider quasifuchsian manifolds with "particles", i.e., cone singularities of fixed angle less than π 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 from the space of quasifuchsian metrics with n particles (of fixed angle) and the product of two copies of the Teichmüller space of a surface with n 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 [BO04] in the non-singular case. RésuméOn considère des variétés quasifuchsiennes "à particules", c'est-à-dire ayant des singularités coniques d'angle fixé inférieur à π allant d'une composante connexe à l'infini à l'autre. Chaque composante connexe du bord à l'infini est alors muni d'une structure conforme marquée par les extrémités des particules. On montre que ceci définit un homéomorphisme de l'espace des métriques quasifuchsiennes à n particules (d'angle fixé) vers le produit de deux copies de l'espace de Teichmüller d'une surface à n points marqués. Ceci étend le théorème de double uniformisation de Bers aux variétés quasifuchsiennes à "particules".Les variétés quasifuchsiennes à particules ont aussi un coeur convexe. Son bord a une métrique induite hyperbolique, avec des singularités coniques aux intersections avec les particules, et est plissé le long d'une lamination géodésique mesurée. On montre que toute paire de métriques hyperboliques à singularités coniques (d'angle prescrit) peut être obtenu, et aussi que toute paire de laminations de plissages, satisfaisant des conditions clairement nécessaires, peut être obtenus, comme dans le cas non-singulier [BO04].

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Section: Definition 14mentioning

confidence: 96%

The convex core of quasifuchsian manifolds with particles

Lecuire

Schlenker

2014

Geom. Topol.

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“…The following theorem, due to Hodgson and Kerckhoff [5] (see also [3]), generalizes Thurston's Dehn filling theorem: …”

Section: Dehn Fillingmentioning

confidence: 96%

Hyperbolic cone-manifolds with large cone-angles

Souto

2003

Geom. Topol.

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“…(When the cone-angle is zero, the tube radius is infinite and the result holds.) The parameterization was extended to geometrically finite cone-manifolds in [Br1].…”

Section: Geometrically Finite Hyperbolic Cone-manifoldsmentioning

confidence: 99%

“…The other properties of v are a straightforward calculation. Next we integrate both sides from 0 to R: By Theorem 4.3 of [Br1] every cohomology class in H 1 (M ; E) that extends to a conformal deformation Φ of the projective boundary is represented by a Hodge form ω. However, it is not shown that ω extends continuously to Φ.…”

Section: Appendix: Mean Value Inequalitiesmentioning

confidence: 99%

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Hyperbolic cone-manifolds, short geodesics, and Schwarzian derivatives

Bromberg

2004

J. Amer. Math. Soc.

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With his hyperbolic Dehn surgery theorem and later the orbifold theorem, Thurston demonstrated the power of using hyperbolic cone-manifolds to understand complete, non-singular hyperbolic 3-manifolds. Hodgson and Kerckhoff introduced analytic techniques to the study of cone-manifolds that they have used to prove deep results about finite volume hyperbolic 3-manifolds. In this paper we use Hodgson and Kerckhoff's techniques to study infinite volume hyperbolic 3-manifolds. The results we will develop have many applications: the Bers density conjecture, the density of cusps on the boundary of quasiconformal deformations spaces, and for constructing type preserving sequences of Kleinian groups.The simplest example of the problem we will study is the following: Let M be a hyperbolic 3-manifold and c a simple closed geodesic in M . Then the topological manifold M \c also has a complete hyperbolic metric which we callM . How does the geometry of M compare to that ofM ? Before attempting to answer such a question, we need to note that if M has infinite volume, the hyperbolic structure will not be unique. If we do not make further restrictions on the choice ofM , then there is no reason to expect that M andM will be geometrically close. If M is convex co-compact, there is a natural choice to make forM . Namely M is compactified by a conformal structure X. We then chooseM to be the unique geometrically finite hyperbolic structure on M \c with conformal boundary X.We can now return to our question: How do the geometry of M andM compare? We will quantify this question in two ways. We will measure the length of geodesics in M andM and we will measure the geometry of the ends of M andM by bounding the distance between the projective structures on their boundaries. What we will see is that the change in geometry is bounded by the length of the geodesic c in the original manifold M .Results of this type were first obtained by McMullen [Mc], in the case of a quasifuchsian manifold, where the geodesic c is also short on a component of the conformal boundary. This work has been extended to arbitrary geometrically finite manifolds by Canary, Culler, Hersonsky and Shalen [CCHS]. Their techniques are entirely different from ours and one goal of this paper is to give new proofs of their estimates. These estimates are a key step in proving the density of cusps in the boundary of quasiconformal deformation spaces of hyperbolic 3-manifolds.

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Rigidity of Geometrically Finite Hyperbolic Cone-Manifolds

Cited by 36 publications

References 15 publications

The convex core of quasifuchsian manifolds with particles

The convex core of quasifuchsian manifolds with particles

Hyperbolic cone-manifolds with large cone-angles

Hyperbolic cone-manifolds, short geodesics, and Schwarzian derivatives

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