Rigidity of inversive distance circle packings revisited

Inversive distance circle packing metric was introduced by P Bowers and K Stephenson [7] as a generalization of Thurston's circle packing metric [34]. They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo [22] proved the infinitesimal rigidity and then Luo [27] proved the global rigidity. In this paper, based on an observation of Zhou [37], we prove this conjecture for inversive distance in (−1,+∞) by variational principles. We also study the global rigidity of a combinatorial curvature introduced in [14], [16], [19] with respect to the inversive distance circle packing metrics where the inversive distance is in (−1,+∞).