On combinatorial geometric flows of two dimensional surfaces

In this note we discuss several versions of discrete Ricci flow on closed two dimensional surfaces. As it was shown by Hamilton and Chow, on a closed surface the Ricci flow converges to the metric of constant curvature for any initial metric. Discrete version of the Ricci flow introduced by Chow and Luo has the same property. This discretization is defined for so called circle packing metrics. We discuss two directions in which results of Chow–Luo are generalized.
On the other hand, straightforward discretization of the Ricci flow on surfaces, which uses a collection of lengths of edges as a metric, for certain initial conditions does not converge to the metric of constant curvature. We give corresponding examples. Moreover, straightforward discretization of the Ricci flow is proved to be equivalent to the combinatorial Yamabe flow on surfaces, introduced by Luo.
Also we discuss generalization of the combinatorial Yamabe flow and its equivalent Ricci flow. In this generalization the vertices of the triangulation are equipped with weights, describing certain inhomogeneity of the surface in response to the tension given by the curvature Based on a large number of numerical experiments, certain conjectures about the behaviour of the solutions of the generalized Yamabe flow are proposed.