On the deformation of inversive distance circle packings, III

Abstract

Given a triangulated surface M, we use Ge–Xu's α-flow [10] to deform any initial inversive distance circle packing metric to a metric with constant α-curvature. More precisely, we prove that the inversive distance circle packing with constant α-curvature is unique if αχ(M)≤0, which generalizes Andreev–Thurston's rigidity results for circle packing with constant cone angles. We further prove that the solution to Ge–Xu's α-flow can always be extended to a solution that exists for all time and converges exponentially fast to constant α-curvature. Finally, we give some combinatorial and topological obstacles for the existence of constant α-curvature metrics.