Rigidity of inversive distance circle packings revisited

Inversive distance circle packing metric was introduced by P Bowers and K Stephenson \cite{BS} as a generalization of Thurston's circle packing metric \cite{T1}. They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo \cite{Guo} proved the infinitesimal rigidity and then Luo \cite{L3} proved the global rigidity. In this paper, based on an observation of Zhou \cite{Z}, we prove this conjecture for inversive distance in $(-1, +\infty)$ by variational principles. We also study the global rigidity of a combinatorial curvature introduced in \cite{GJ4,GX4,GX6} with respect to the inversive distance circle packing metrics where the inversive distance is in $(-1, +\infty)$.