Convergences of combinatorial Ricci flows to degenerated circle packings in hyperbolic background geometry

Abstract

This paper investigates a kind of degenerated circle packings in hyperbolic background geometry. A main problem is whether a prescribed total geodesic curvature data can be realized by a degenerated circle packing or not. We fully characterize the sufficient and necessary conditions and show the uniqueness. Furthermore, we introduce the combinatorial Ricci flow to find the desired degenerated circle packed surface, analogous to the methods of Chow-Luo [7] and Takatsu [37].