Seminario DAGA: The Plateau problem, Fuchsian systems and the Riemann-Hilbert problem

Venerdi 15 Ottobre 2010 - Laura Desideri


Venerdi 15 Ottobre 2010, ore 14:30 in Aula 2BC60, Laura Desideri (Tubinga) terra' un seminario dal titolo "The Plateau problem, Fuchsian systems and the Riemann-Hilbert problem".

The Plateau problem is to prove than any given closed and connected Jordan curve in R3 bounds at least one minimal surface of disk-type. The first almost complete resolutions are given in the early 1930's by Douglas and Rado. However, in 1928, Garnier published a resolution for polygonal boundary curves which seems to have been forgotten. His proof is really different from the variational method, it relies on the fact that one can associate with each minimal disk with a polygonal boundary curve a real Fuchsian second-order equation defined on the Riemann sphere. The monodromy of the equation is determined by the oriented directions of the edges of the boundary. To solve the Plateau problem, we are thus led to solve a Riemann-Hilbert problem and to use isomonodromic deformations of Fuchsian equations. I will give a sketch of the proof, and I will briefly explain how Garnier's result can be extended into Minkowski 3-space.

Rif. int. A. Bertapelle