Tame and relatively elliptic ℂℙ1–structures onthe thrice-punctured sphere

Suppose a relatively elliptic representation of the fundamental group of the thrice-punctured sphere S is given.We prove that all projective structures on S with holonomy 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 .In the process, we show that (on a general surface of negative Euler characteristics) structures satisfying these conditions can be characterized in terms of their Mbius completion, and in terms of certain meromorphic quadratic differentials.30F30, 57M50 1. Introduction 4589 2. Basics on complex projective geometry 4596 3. Tame and relatively elliptic CP 1 -structures 4612 4. The complex analytic point of view 4632 5.