## Discrete geometric flows on two-dimensional compact oriented surfacesдоклад на конференции

• Автор:
• Международная Конференция : XX Geometrical Seminar
• Даты проведения конференции: 20-24 мая 2018
• Дата доклада: 21 мая 2018
• Тип доклада: Устный
• Докладчик: не указан
• Место проведения: Vrnjacka Banja, Serbia
• Аннотация доклада:

Geometric flows, or differential equations for families of time-dependent metrics, have been investigated in mathematics for a very long time. The Ricci flow and the mean curvature flow are the most commonly used examples. Perelman's proof of Thurston's conjecture on the geometrization of three-dimensional manifolds and, in particular, the Poincare conjecture, is considered to be the most effective application of Ricci flows. Theorems on the convergence of the Ricci flow on two-dimensional closed oriented surfaces to the metric of constant curvature were among the first to be proved. Currently several approaches to the discretization of Ricci flows exist. This work discusses the most natural way of discretization by the Ricci flows on two-dimensional surfaces proposed by Chow and Luo. In their approach the Thurston’s circle packing metric is considered as a metric on triangulated surface. Later it has been demonstrated that the Ricci flow on surfaces in the combinatorial setting converges to the Thruston circle packing metric. The combinatorial structure of a triangulated surface also includes a set of weights on all its edges. A crucial assumption for the convergence of the flow in was the condition of non-negativity of these weights. This paper demonstrates that the theorem of convergence of the Ricci flow to a unique metric of constant curvature can be proved under weakened conditions on the weights. Examples of weight sets on the edges of a triangulated surface allowing the existence of several metrics of constant curvature, are also demonstrated. Moreover, a numerical simulation of the flow in the vicinity of these solutions is presented. The discretization of the mean curvature flow on the surface of revolution, given by a special partition of the icosahedron, is also demonstrated. The numerical solution demonstrates the formation of singularities by the mean curvature flow for the case when the surface does not satisfy the conditions of Husiken's theorem.

• Добавил в систему: Пепа Руслан Юрьевич