A combinatorial curvature flow in spherical background geometry

Abstract

In [12], the existence of ideal circle patterns in Euclidean or hyperbolic background geometry under combinatorial conditions was proved using flow approaches. It remains as an open problem for the spherical case. In this paper, we introduce a combinatorial geodesic curvature flow in spherical background geometry, which is analogous to the combinatorial Ricci flow of Chow and Luo in [4]. We characterize the sufficient and necessary condition for the convergence of the flow. That is, the prescribed geodesic curvature satisfies certain geometric and combinatorial condition if and only if for any initial data the flow converges exponentially fast to a circle pattern with given total geodesic curvature on each circle. Our result could be regarded as a resolution of the problem in the spherical case. As far as we know, this is the first combinatorial curvature flow in spherical background geometry with fine properties, and it provides an algorithm to find the desired ideal circle pattern.