Abstract:
After proving the convergence of the Ricci flow on a two-dimensional closed
surface for any initial data to the metric of constant curvature, the question
of discretization of the flow naturally arose.
Some conceptual difficulty is related to the fact that the metric on a
triangulated surface
is determined by the lengths of the edges, while the curvature is concentrated
at the vertices.
The naive version of the Ricci flow, as we will see, does not satisfy the
desired property of convergence of the flow to the metric of constant
curvature for any initial metric. A positive solution was found in a class of
circle packing metrics, which itself is a very interesting combinatorial
object. We will discuss the corresponding combinatorial Ricci flow, as well as
some of its generalizations.
