Circle packings on surfaces with projective structures and uniformization

Let Σ_g be a closed orientable surface of genus g \geq 2 and τa graph on Σ_g with one vertex which lifts to a triangulation of the universal cover. We have shown that the cross ratio parameter space \mathcal{C}_τassociated 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 \mathbb{R}^{6g-6}, and moreover that the forgetting map of \mathcal{C}_τto 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}_τto the Teichmüller space \mathcal{T}_g is proper.

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