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Combinatorial Calabi flows are introduced by Ge in his Ph.D. thesis(Combinatorial methods and geometric equations, Peking University, Beijing,2012), and have been studied extensively in Euclidean and hyperbolic backgroundgeometry. In this paper, we introduce the combinatorial Calabi flow inspherical background geometry for finding ideal circle patterns with prescribedtotal geodesic curvatures. We prove that the solution of combinatorial Calabiflow exists for all time and converges if and only if there exists an idealcircle pattern with prescribed total geodesic curvatures. We also show that ifit converges, it will converge exponentially fast to the desired metric, whichprovides an effective algorithm to find certain ideal circle patterns. To ourknowledge, it is the first combinatorial Calabi flow in spherical backgroundgeometry.

Combinatorial Calabi flows for ideal circle patterns in spherical background geometry