Circle packings on surfaces with projective structures and uniformization

Let \Sigma_g be a closed orientable surface of genus g \geq 2 and \tau a graph on \Sigma_g with one vertex which lifts to a triangulation of the universal cover. We have shown that the cross ratio parameter space $\mathcal{C}$\tau associated with \tau, which can be identified with the set of all pairs of a projective structure and a circle packing on it with nerve isotopic to \tau, is homeomorphic to $\mathbb{R}$^{6g-6}, and moreover that the forgetting map of $\mathcal{C}$\tau to the space of projective structures is injective. In this paper, we show that the composition of the forgetting map with the uniformization from $\mathcal{C}$_\tau to the Teichm"uller space $\mathcal{T}$_g is proper.