Branched projective structures with quasi-Fuchsian holonomy

TL;DR

This paper proves that for a closed surface with negative Euler characteristic and a quasi-Fuchsian holonomy, the space of branched projective structures with fixed holonomy and branch points is connected, and describes its smooth structure.

Contribution

It establishes the connectedness of the deformation space of branched projective structures with quasi-Fuchsian holonomy and provides an explicit smooth complex manifold structure for this space.

Findings

Deformation space M(k,R) is connected for k>0.

Two structures with same holonomy and branch points are related by moving branch points.

M(k,R) is a smooth complex manifold modeled on Hurwitz spaces.

Abstract

We prove that if S is a closed compact surface of negative Euler characteristic, and if R is a quasi-Fuchsian representation in PSL(2,C), then the deformation space M(k,R) of branched projective structures on S with total branching order k and holonomy R is connected, as soon as k>0. Equivalently, two branched projective structures with the same quasi-Fuchsian holonomy and the same number of branch points are related by a movement of branch points. In particular grafting annuli are obtained by moving branch points. In the appendix we give an explicit atlas for the space M(k,R). It is shown to be a smooth complex manifold modeled on Hurwitz spaces.