The convex core of quasifuchsian manifolds with particles

TL;DR

This paper extends the classical theory of quasifuchsian manifolds by incorporating cone singularities called particles, establishing a homeomorphism between their moduli space and Teichmüller space, and analyzing their convex core geometry.

Contribution

It introduces a framework for quasifuchsian manifolds with particles, proving a homeomorphism with Teichmüller space and characterizing their convex core boundary geometry.

Findings

Homeomorphism between space of quasifuchsian metrics with particles and Teichmüller space

Existence of convex cores with prescribed cone singularities and bending laminations

Extension of classical quasifuchsian theory to include cone singularities

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 Teichm\"uller 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.