Combinatorial curvature flows for generalized hyperbolic circle packings on surfaces

Generalized hyperbolic circle packings were introduced in Ba et al. (2023) as the generalization of tangential circle packings in hyperbolic background geometry. To find generalized hyperbolic circle packings on surfaces with prescribed total geodesic curvatures, we introduce the combinatorial Calabi flow, the fractional combinatorial Calabi flow and the combinatorial $p$th Calabi flow for generalized hyperbolic circle packings on surfaces. We establish several equivalent conditions regarding the longtime behaviors of these combinatorial curvature flows. This provides effective algorithms for finding the generalized hyperbolic circle packings with prescribed total geodesic curvatures on surfaces.