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A Schlaeflitype formula for the convex core of hyperbolic 3manifolds
 J.Diff.Geom
, 1998
"... Let M be a (connected) hyperbolic 3–manifold, namely a complete Riemannian manifold of dimension 3 and of constant sectional curvature −1, with finitely generated fundamental group. A fundamental subset of M is its convex core CM, which is the smallest nonempty convex subset of M. The condition tha ..."
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Cited by 19 (1 self)
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Let M be a (connected) hyperbolic 3–manifold, namely a complete Riemannian manifold of dimension 3 and of constant sectional curvature −1, with finitely generated fundamental group. A fundamental subset of M is its convex core CM, which is the smallest nonempty convex subset of M. The condition that the volume of CM is finite is open in the space of hyperbolic metrics on M, provided we restrict attention to cusprespecting deformations. In this paper, we give a formula which, for a cusppreserving variation of the hyperbolic metric of M, expresses the variation of the volume of the convex core CM in terms of the variation of the bending measure of its boundary. This formula is analogous to the Schläfli formula for the volume of an n–dimensional hyperbolic polyhedron P; see [Sc1][Kne][AVS] and §1. If the metric of P varies, the Schläfli formula expresses the variation of the volume of P in terms of the variation of the dihedral angles of P along the (n − 2)–faces of its boundary and of the (n − 2)–volumes of these faces. The analogy stems from the fact that the boundary ∂CM of CM is almost polyhedral, in the sense that it is totally geodesic almost everywhere. However, the pleating locus, where ∂CM is not totally geodesic, is not a finite collection of edges any more. Typically, it will consist of uncountably many infinite geodesics. In addition, the topology of this pleating locus can drastically change as we vary the metric of M. So the
Hyperbolic manifolds with polyhedral boundary
, 2001
"... Let (M, ∂M) be a compact 3manifold with boundary which admits a complete, convex cocompact hyperbolic metric. We are interested in the following question: Question A Let h be a (nonsmooth) metric on ∂M, with curvature K> −1. Is there a unique hyperbolic metric g on M, with convex boundary, such t ..."
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Cited by 11 (5 self)
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Let (M, ∂M) be a compact 3manifold with boundary which admits a complete, convex cocompact hyperbolic metric. We are interested in the following question: Question A Let h be a (nonsmooth) metric on ∂M, with curvature K> −1. Is there a unique hyperbolic metric g on M, with convex boundary, such that the induced metric on ∂M is h? There is also a dual statement: Question B Let h be a (nonsmooth) metric on ∂M, such that its universal cover is CAT(1). Is there a unique hyperbolic metric g on M, with convex boundary, such that the third fundamental form of ∂M is h? Many partial results are known on those questions when M is a ball, and a few in more general cases. We are interested here in the special case where ∂M locally looks like an ideal polyhedron in H 3. We can give a fairly complete answer to question B – which in this case concerns the dihedral angles – and some partial results on question A. This also has some interesting byproducts, for instance a flat affine structure on the Teichmüller space of a surface with some marked points. Résumé
The WeilPetersson and Thurston Symplectic Forms
 DUKE MATH. J
"... We consider the WeilPetersson form on the Teichmüller space T(S) of a surface S of genus at least 2, and compute it in terms of the shearing coordinates for T(S) associated to a geodesic lamination # on S. In the corresponding expression, the WeilPetersson form coincides with Thurston's intersec ..."
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Cited by 8 (0 self)
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We consider the WeilPetersson form on the Teichmüller space T(S) of a surface S of genus at least 2, and compute it in terms of the shearing coordinates for T(S) associated to a geodesic lamination # on S. In the corresponding expression, the WeilPetersson form coincides with Thurston's intersection form on the space of transverse cocycles for #.
A brief survey of the deformation theory of Kleinian Groups
 GEOMETRY & TOPOLOGY MONOGRAPHS VOLUME 1: THE EPSTEIN BIRTHDAY SCHRIFT PAGES 23–49
"... ..."
Projective structures, grafting, and measured laminations. Geometry and Topology
, 2008
"... We show that grafting any fixed hyperbolic surface defines a homeomorphism from the space of measured laminations to Teichmüller space, complementing a result of Scannell–Wolf on grafting by a fixed lamination. This result is used to study the relationship between the complexanalytic and geometric ..."
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Cited by 5 (1 self)
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We show that grafting any fixed hyperbolic surface defines a homeomorphism from the space of measured laminations to Teichmüller space, complementing a result of Scannell–Wolf on grafting by a fixed lamination. This result is used to study the relationship between the complexanalytic and geometric coordinate systems for the space of complex projective (CP 1) structures on a surface. We also study the rays in Teichmüller space associated to the grafting coordinates, obtaining estimates for extremal and hyperbolic length functions and their derivatives along these grafting rays. 30F60; 30F10, 30F40, 32G15, 57M50 1
Hyperbolic manifolds with . . .
, 2002
"... Let (M, ∂M) be a compact 3manifold with boundary which admits a complete, convex cocompact hyperbolic metric. Thurston asked whether, for any smooth metric h on ∂M with curvature K> −1, there is a unique hyperbolic metric g on M, with convex boundary, inducing h on ∂M. We consider the dual questio ..."
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Let (M, ∂M) be a compact 3manifold with boundary which admits a complete, convex cocompact hyperbolic metric. Thurston asked whether, for any smooth metric h on ∂M with curvature K> −1, there is a unique hyperbolic metric g on M, with convex boundary, inducing h on ∂M. We consider the dual question: Question. Let h be a (nonsmooth) metric on ∂M, such that its universal cover is CAT(1). Is there a unique hyperbolic metric g on M, with convex boundary, such that the third fundamental form of ∂M is h? We are interested here in the special case where ∂M locally looks like an ideal polyhedron in H 3. We can give a fairly complete answer to the question on the third fundamental form — which in this case concerns the dihedral angles — and some partial results about the induced metric. This has some byproducts, like a flat affine structure on the Teichmüller space of a surface with some marked points.
The WeilPetersson metric and the renormalized volume of hyperbolic 3manifolds
, 907
"... Abstract. We survey the renormalized volume of hyperbolic 3manifolds, as a tool for Teichmüller theory, using simple differential geometry arguments to recover results sometimes first achieved by other means. One such application is McMullen’s quasifuchsian (or more generally Kleinian) reciprocity, ..."
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Abstract. We survey the renormalized volume of hyperbolic 3manifolds, as a tool for Teichmüller theory, using simple differential geometry arguments to recover results sometimes first achieved by other means. One such application is McMullen’s quasifuchsian (or more generally Kleinian) reciprocity, for which different arguments are proposed. Another is the fact that the renormalized volume of quasifuchsian (or more generally geometrically finite) hyperbolic 3manifolds provides a Kähler potential for the WeilPetersson metric on Teichmüller space. Yet another is the fact that the grafting map is symplectic, which is proved using a variant of the renormalized volume defined for hyperbolic ends. Contents
Geometry & Topology Monographs Volume 1: The Epstein Birthday Schrift Pages 23–49
"... A brief survey of the deformation theory ..."