The character of Thurston’s circle packings

We introduce the character of Thurston’s circle packings in the hyperbolic background geometry Consequently, some quite simple criteria are obtained for the existence of hyperbolic circle packings. For example, if a closed surface X admits a circle packing with all the vertex degrees d_i ⩾ 7, then it admits a unique complete hyperbolic metric so that the triangulation graph of the circle packing is isotopic to a geometric decomposition of X. This criterion is sharp due to the fact that any closed hyperbolic surface admits no triangulations with all d_i ⩽ 6. As a corollary, we obtain a new proof of the uniformization theorem for closed surfaces with genus g ⩾ 2, and moreover, any hyperbolic closed surface has a geometric decomposition. To obtain our results, we use Chow-Luo’s combinatorial Ricci flow as a fundamental tool.