Tame and relatively elliptic CP¹ –structures on the thrice-punctured sphere

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 † 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.