Tame and relatively elliptic $\mathbb{CP}^1$-structures on the thrice-punctured sphere

TL;DR

This paper characterizes tame, relatively elliptic $ ext{CP}^1$-structures on the thrice-punctured sphere via grafting circular triangles, linking them to holonomy representations and meromorphic quadratic differentials.

Contribution

It provides a new construction method for such structures using grafting of circular triangles determined by a natural framing of the holonomy.

Findings

Structures can be obtained by grafting circular triangles.

Characterization via M"obius completion and meromorphic quadratic differentials.

Applicable to general surfaces with negative Euler characteristic.

Abstract

Suppose a relatively elliptic representation $\rho$ of the fundamental group of the thrice-punctured sphere $S$ is given. We prove that all projective structures on $S$ with holonomy $\rho$ and satisfying a tameness condition at the punctures can be obtained by grafting certain circular triangles. The specific collection of triangles is determined by a natural framing of $\rho$. In the process, we show that (on a general surface $\Sigma$ of negative Euler characteristics) structures satisfying these conditions can be characterized in terms of their M\"obius completion, and in terms of certain meromorphic quadratic differentials.