Details

Autor(en) / Beteiligte

• Pepa, R. Yu
• Popelensky, Th. Yu

Titel

On Combinatorial Geometric Flows of Two Dimensional Surfaces

Ist Teil von

• Russian mathematics, 2021-05, Vol.65 (5), p.60-68

Ort / Verlag

Moscow: Pleiades Publishing

Erscheinungsjahr

2021

Quelle

SpringerNature Journals

Beschreibungen/Notizen

• 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.