Variations of the Boundary Geometry of 3{dimensional Hyperbolic Convex Cores

Let M be a hyperbolic 3–manifold, namely a complete 3–dimensional Riemannian manifold of constant curvature −1, such that the fundamental group π1 (M) is finitely generated. A fundamental subset of M is its convex core CM , defined as the smallest non-empty closed convex subset ofM . The boundary ∂CM of this convex core is a surface of finite topological type, and its geometry was described by W.P. Thurston [Thu]: The surface ∂CM is almost everywhere totally geodesic, and is bent along a family of disjoint geodesics called its pleating locus . The path metric induced by the metric of M is hyperbolic, and the way ∂CM is bent is completely determined by a certain measured geodesic lamination. We want to investigate how the geometry of ∂CM varies as we deform the metric ofM . For technical reasons, in particular because we do not want the topology of ∂CM to change, we choose to restrict attention to quasi-isometric deformations ofM , namely hyperbolic manifoldsM ′ for which there exists a diffeomorphism M → M ′ whose differential is uniformly bounded. In the language of Kleinian groups, a quasi-isometric deformation of M is also equivalent to a quasi-conformal deformation of its holonomy; see [Thu, §10]. This is not a very strong restriction. For instance, in the conjecturally generic case whereM is geometrically finite without cusps, every small deformation of the metric is quasi-isometric. When M is geometrically finite, quasi-isometric deformations of the metric coincide with deformations of the holonomy π1 (M) → Isom ( H )