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

Abstract

Suppose a relatively elliptic representation ρ of the fundamental group of the thrice-punctured sphere S is given. We prove that all projective structures on S with holonomy ρ 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 ρ. In the process, we show that (on a general surface Σ of negative Euler characteristics) structures satisfying these conditions can be characterized in terms of their Möbius completion, and in terms of certain meromorphic quadratic differentials. Contents 1. Introduction 1 2. Basics on complex projective geometry 7 2.1. Configurations of circles 7 2.2. Elliptic Möbius transformations 9 2.3. Triangular immersions 12 3. Tame and relatively elliptic CP 1 -structures 21 3.1. Ends, framing, and grafting 22 3.2. The Möbius completion 26 3.3. Local properties of the developing map at an end 33 3.4. The index of a puncture 38 4. The complex analytic point of view 39 4.1. Local theory at regular singularities 39 4.2. Meromorphic projective structures 41 5. Structures on the thrice-punctured sphere 43 5.1. Triangular structures 44 5.2. Grafting Theorems A and B 49 Appendix A. Tables of atomic triangular immersions 53 References 56