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 nonuniqueness coming from Möbius transformations. In this paper, we show that a unique existence result for circle pattern with spherical conical metric holds if one prescribes the total geodesic curvature of each circle instead of the cone angles.