Problème de Plateau, équations fuchsiennes et problème de Riemann-Hilbert

Details Problème de Plateau, équations fuchsiennes et problème de Riemann-Hilbert Problème de Plateau, équations fuchsiennes et problème de Riemann-Hilbert

2010-03-04

arxiv.org/abs/1003.0978v2

Mathematics

Differential Geometry

French, 106 pages

Scientific paper

This dissertation is devoted to the resolution of the Plateau problem in the case of polygonal boundary curves in three-dimensional Euclidean space. It relies on the method developed by Ren\'e Garnier and published in 1928 in a paper which seems today to be totally forgotten. Garnier's approach is more geometrical and constructive than the variational one, and it provides minimal disks without branch point. However, it is sometimes really complicated, and even obscure or incomplete. Following Garnier's initial ideas, we propose a new proof, which intends not only to be complete, but also simpler and moderner than his one. This work mainly relies on a systematic use of Fuchsian systems and on the relation that we establish between the reality of such systems and their monodromy. Garnier's method is based on the following fact: using the spinor Weierstrass representation for minimal surfaces, we can associate a real Fuchsian second-order equation, defined on the Riemann sphere, with each minimal disk with a polygonal boundary curve. 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. We then proceed in two steps: first, by means of isomonodromic deformations, we construct and describe the family of all minimal disks with a polygonal boundary curve of given oriented directions. Then we use this description to study the edges's lengths of their boundary curves, and we show that every polygon is the boundary of a minimal disk.

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