In this paper, 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 a constant curvature for any initial metric. The 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, the direct discretization of the Ricci flow on surfaces, which uses a collection of edges lengths as a metric, does not converge to the metric of constant curvature for certain initial conditions. We give the corresponding examples. Moreover, the direct discretization of the Ricci flow is proved to be equivalent to the combinatorial Yamabe flow on surfaces introduced by Luo. In addition, we discuss a 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 inhomogeneities of the surface in response to the tension given by the curvature to the metric. Based on a large number of numerical experiments, certain conjectures about the behavior of the solutions of the generalized Yamabe flow are proposed.

在本文中，我们讨论了封闭二维表面上离散 Ricci 流的几种版本。正如 Hamilton 和 Chow 所表明的那样，在闭合曲面上，对于任何初始度量，Ricci 流都收敛到恒定曲率的度量。Chow 和 Luo 引入的离散版本的 Ricci 流具有相同的性质。这种离散化是为所谓的圆包装度量定义的。我们讨论了 Chow-Luo 的结果推广的两个方向。另一方面，表面上 Ricci 流的直接离散化（使用边长的集合作为度量）在某些初始条件下不会收敛到恒定曲率的度量。我们给出相应的例子。而且，证明 Ricci 流的直接离散化等效于由 Luo 引入的表面上的组合 Yamabe 流。此外，我们讨论了组合 Yamabe 流及其等效 Ricci 流的推广。在这个概括中，三角剖分的顶点配备了权重，描述了表面的某些不均匀性，以响应由曲率给度量的张力。基于大量的数值实验，提出了关于广义Yamabe流解的行为的某些猜想。描述表面的某些不均匀性，以响应由曲率给度量的张力。基于大量的数值实验，提出了关于广义Yamabe流解的行为的某些猜想。描述表面的某些不均匀性，以响应由曲率给度量的张力。基于大量的数值实验，提出了关于广义Yamabe流解的行为的某些猜想。