Заједнички састанак Одељења за математику и Одељења за механику, 8. мај 2017.

• 08. Мај, 2017
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Заједнички састанак Одељења за математику и Одељења за механику биће одржан у понедељак, 8. маја 2017. у сали 301ф Математичког института САНУ са почетком у 14:15 часова.

Предавач: Teodor Popelenski, MGU, Moscow

Наслов предавања: ON COMBINATORIAL RICCI FLOW ON SURFACES

Апстракт: After R.Hamilton`s paper (1982) Three-manifolds with positive Ricci curvature natural question the natural question about properties of the Ricci flow on surfaces have arisen. In this dimension the long-time existence and convergence were proved more or less easy: in 1986 R.Hamilton announced and in 1988 published the proof of convergence of the Ricci flow to the metric of constant curvature for arbitrary initial metric for any closed surface different form the sphere; in 1991 B.Chow closed the question by proving the same statement for two-dimensional sphere.

In 2003 B.Chow and F.Luo investigated on of possible "discretization" of the Ricci flow. Fixed data consists of a closed surface, its triangulation, and weights on the edges of the triangulation. For this object one has so called "circle packing metrics", corresponding curvatures, and Ricci flow. This version of discretization is important due to the circle packing which were investigated by Thurston in his unpublished book "Geometry and topology of 3-manifolds."

Chow and Luo showed that under certain conditions on the weight function the Ricci flow converges exponentially fast to the metric of constant curvature. One of the important conditions consists in non-negativity of the weights.

Recently R.Pepa and me were able to weaken some of the Clow-Luo conditions. Namely some weights can be negative but still should satisfy some conditions. Also we show that the weakening the conditions cannot be unlimited. We found examples of triangulation of surfaces and weights on the edges of the triangulation such that there exists saddle points of the Ricci flow.

In the talk I give the exposition of old results and present some new ones.