Circle packings and total geodesic curvatures in hyperbolic background geometry

Abstract: In this paper, we study a new type of circle packings in hyperbolic background geometry. Horocycles and hypercycles are also considered in this packing. We give the existence and rigidity of this type of circle packing with conical singularities in terms of the total geodesic curvature. Moreover, we introduce the combinatorial curvature flow on surfaces to find the desired circle packing with the prescribed total geodesic curvature.