The convex core of quasifuchsian manifolds with particles

Abstract

We consider quasifuchsian manifolds with "particles", i.e., cone singularities of fixed angle less than $\pi$ 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 is analoguous to the Bers theorem in the non-singular case. 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 [BO] in the non-singular case.

38 pages, 1 figure. v2: small corrections and improved exposition, 43 pages

References in corpus (7)

• On the renormalized volume of hyperbolic 3-manifolds
• Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography
• Fixed points of compositions of earthquakes
• Hyperbolic manifolds with convex boundary
• Collisions of particles in locally AdS spacetimes I. Local description and global examples
• Collars and partitions of hyperbolic cone-surfaces
• A symplectic map between hyperbolic and complex Teichmüller theory