On Thurston's parametrization of $\mathbb{C}{\rm P}^1$-structures

Abstract

Thurston related $\mathbb{C}{\rm P}^1$-structures (complex projective structures) and equivariant pleated surfaces in the hyperbolic-three space $\mathbb{H}^3$, in order to give a parameterization of the deformation space of $\mathbb{C}{\rm P}^1$-structures. In this note, we summarize Thurston's parametrization of $\mathbb{C}{\rm P}^1$-structures, based on Kamishima-Tan and Kulkani-Pinkall. We, in addition, give independent proofs for the following well-known theorems on $\mathbb{C}{\rm P}^1$-structures by means of pleated surfaces given by the parameterization. (1) Goldman's Theorem on $\mathbb{C}{\rm P}^1$-structures with quasi-Fuchsian holonomy. (2) The path lifting property of developing maps in their domains of discontinuity in $\mathbb{C}{\rm P}^1$.