The famous Koebe-Andreev-Thurston Theorem for Euclidean and hyperbolic circle packings has been proved for many years. Bobenko and Springborn further generalized the Koebe-Andreev-Thurston Theorem to Euclidean and hyperbolic circle patterns on surfaces. Due to the nonconvexity of the classical energy function for spherical circle patterns, a similar result fails for the spherical circle packings on surfaces if one wants to prescribe the cone angle at the vertex. It has been a longstanding open problem that whether there exist similar results for spherical circle patterns on surfaces.
In this paper, the author cleverly introduces a new combinatorial curvature for spherical circle patterns on surfaces, which is called the total geodesic curvature. For the center of a circle in the circle pattern, the total geodesic curvature is defined to be the total geodesic curvature of the circle in the spherical background geometry. Using this total geodesic curvature, the author establishes an elegant unique existence result, Theorem 1, for spherical circle patterns on surfaces, whose statement is almost identical to Bobenko-Springborn’s generalization of Koebe-Andreev-Thurston Theorem. This result can be taken as a spherical version of the famous Koebe-Andreev-Thurston Theorem, and solves the longstanding open problem mentioned above.