Hyperbolic Circle Packings and Total Geodesic Curvatures on Surfaces with Boundary

This paper investigates a generalized hyperbolic circle packing (including circles, horocycles or hypercycles) with respect to the total geodesic curvatures on the surface with boundary. We mainly focus on the existence and rigidity of circle packing whose contact graph is the $1$-skeleton of a finite polygonal cellular decomposition, which is analogous to the construction of Bobenko and Springborn [4]. Motivated by Colin de Verdi\`ere's method [6], we introduce the variational principle for generalized hyperbolic circle packings on polygons. By analyzing limit behaviours of generalized circle packings on polygons, we give an existence and rigidity for the generalized hyperbolic circle packing with conical singularities regarding the total geodesic curvature on each vertex of the contact graph. As a consequence, we introduce the combinatoral Ricci flow to find a desired circle packing with a prescribed total geodesic curvature on each vertex of the contact graph.