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

## TL;DR

The paper demonstrates that complex projective structures with quasi-Fuchsian holonomy can be transformed through a minimal sequence of bubbling and debubbling, connecting them to uniformizing structures.

## Contribution

It introduces a simple method to perform simultaneous (de)grafting of such structures using just one bubbling and one debubbling step.

## Key findings

- Any structure with quasi-Fuchsian holonomy can be connected to the uniformizing structure via a minimal sequence.
- The process involves a single bubbling and debubbling step, simplifying previous methods.
- The approach provides a new perspective on the structure space of complex projective structures.

## 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 $\rho:\pi_1(S)\to$ PSL$_2\mathbb{C}$ can be joined to the corresponding uniformizing structure $\sigma_\rho$ by a simple sequence of one bubbling and one debubbling, with a stopover in the space of branched complex projective structures.