Multi(de)grafting quasi-Fuchsian complex projective structures via bubbles

Abstract

We show that the simultaneous (de)grafting of a complex projective structure with quasi-Fuchsian holonomy along a multicurve can be performed by a simple sequence of one bubbling and one debubbling. As a consequence we obtain that any complex projective structure with quasi-Fuchsian holonomy ρ:π1(S)→ PSLC2 can be joined to the corresponding uniformizing hyperbolic structure σρ by a simple sequence of one bubbling and one debubbling, with a stopover in the space of branched complex projective structures.