Circle packings on surfaces with projective structures

The Koebe-Andreev-Thurston theorem states that for any triangulation of a closed orientable surface Sigma(g) of genus g which is covered by a simple graph in the universal cover, there exists a unique metric of curvature 1, 0 or -1 on the surface depending on whether g = 0, 1 or greater than or equal to 2 such that the surface with this metric admits a circle packing with combinations given by the,triangulation. Furthermore, the circle packing is essentially rigid, that is, unique up to conformal automorphisms of the surface isotopic to the identity. In this paper, we consider projective structures on the surface where circle packings are also defined. We show that the space of projective structures on a surface of genus g greater than or equal to 2 which admits a circle packing contains a neighborhood of the Koebe-Andreev-Thurston structure homeomorphic to R6g-6. We furthermore show that if a circle packing consists of one circle, then the space is globally homeomorphic to R6g-6 and that the circle packing is rigid.