On circle patterns and spherical conical metrics

Abstract: The Koebe-Andreev-Thurston circle packing theorem, as well as its generalization to circle patterns due to Bobenko and Springborn, holds for Euclidean and hyperbolic metrics possibly with conical singularities, but fails for spherical metrics because of the non-uniqueness coming from M\"obius transformations. In this paper, we show that a unique existence result for circle pattern with spherical conical metric holds if one prescribes the geodesic total curvature of each circle instead of the cone angles.