Circle packings on surfaces with projective structures and uniformization

Let Σ_g be a closed orientable surface of genus g ≥ 2 and a graph on Σ_g with one vertex that lifts to a triangulation of the universal cover. We have shown before that the cross ratio parameter space C{script}_τ associated with τ, which can be identified with the set of all pairs of a projective structure and a circle packing on it with nerve isotopic to τ is homeomorphic to R{double-struck}^6g-6, and moreover that the forgetting map of C{script}_τ to the space of projective structures is injective. Here we show that the composition of the forgetting map with the uniformization from C{script}_τ to the Teichmüller space T{script}_g is proper.