A generalized combinatorial Ricci flow on surfaces of finite topological type

Abstract

We introduce a generalized combinatorial Ricci flow on surfaces of finite topological type. Using a Lyapunov function, we prove that the flow exists for all time and converges to a circle pattern metric on surfaces with prescribed curvatures. This suggests an algorithm to find circle patterns on surfaces with obtuse exterior intersection angles. As a comparison, this flow has the advantage of accelerating the convergence rate.