Combinatorial p-th Geodesic Curvature Flows for Spherical Ideal Circle Patterns

We introduce a combinatorial p-th geodesic curvature flow for ideal circle patterns in spherical background geometry, where p>1. We prove that, for any initial data, the flow has a unique solution for all time. Moreover, prescribed total geodesic curvatures can be realized by a spherical ideal circle pattern if and only if the solution of the flow converges for some initial data, or equivalently, for any initial data. In the realizable case, the solution converges in finite time for 1<p<2, exponentially fast for p=2, and at an algebraic rate for p>2. For a constrained flow preserving the sum of the logarithmic curvatures, we characterize convergence to a uniformly shifted target and show that the preserved sum determines the limiting metric. Finally, we characterize the admissible targets by positive edge allocations and obtain a vertexwise sufficient condition for existence.