Combinatorial Ricci flow and Calabi flow for generalized hyperbolic circle packings

In Hu et al. (2025), they studied a generalized hyperbolic circle packing (including circles, horocycles and hypercycles) with a total geodesic curvature on each generalized circle of this circle packing and a discrete Gaussian curvature on the center of each dual circle. In this paper, we introduce the combinatorial Ricci flow and combinatorial Calabi flow to find this type of generalized circle packings for a data including prescribed total geodesic curvatures of generalize circles and discrete Gaussian curvatures on centers of dual disks. We show that the solution to the combinatorial Ricci flow and combinatorial Calabi flow in the hyperbolic geometry with the given initial value exists for all the time and converges exponentially fast to a unique generalized circle packing metric.