Combinatorial pth Calabi flows for total geodesic curvatures in spherical background geometry

The concept of combinatorial pth Calabi flow is proposed for finding circle packing metrics with prescribed curvatures. It has been studied under the Euclidean and hyperbolic background geometries. This study investigates the combinatorial pth Calabi flow for total geodesic curvatures under the spherical background geometry. It is established that the combinatorial pth Calabi flow, subject to given total geodesic curvatures under the spherical background geometry, exists for all time t ∈ [0, + ∞) and converges to an ideal circle pattern metric. This finding introduces a new algorithm for constructing ideal circle pattern metrics with prescribed total geodesic curvatures.