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Seminários do IMPA

Geometria Diferencial

Discrete geometry of surfaces towards the filling area conjecture
Marcos Cossarini

Terça-feira, 10 de abril de 2018, 15:30
Sala 236.

In 1983, Gromov conjectured that the hemisphere has smallest area among Riemannian surfaces that fill isometrically a circle of given length. We discuss the history of the problem and present a discrete version, where a cycle graph of length 2n is filled isometrically by a combinatorial square-celled surface with the least possible number of cells, conjectured to be n(n-1)/2. We show that our discrete problem is equivalent to the continuous problem, as extended by Ivanov to admit filling surfaces with self-reverse Finsler metrics, because every such continuous metric can be replaced by a square-celling whose lengths and areas approximate the original values as precisely as desired (quasi-isometrically with arbitrarily good additive and multiplicative constants).

In a second talk we attempt to discretize Finsler metrics that are not self-reverse. For that purpose we introduce on the surface a discrete structure called "fine", which consists of an embedded directed graph that divides the surface into triangular cells, and where moving along and edge costs one or zero according to whether we go in one way or the reverse. Fine structures conjecturally approximate any Finsler metric on a surface, and may extend to higher dimensions as well.