Probl\`eme de Plateau, \'equations fuchsiennes et probl\`eme de Riemann-Hilbert

TL;DR

This dissertation presents a new, simplified proof for solving the Plateau problem with polygonal boundaries in 3D space, utilizing Fuchsian systems and Riemann-Hilbert problems to construct minimal disks without branch points.

Contribution

It offers a modern, complete alternative to Garnier's geometric method by systematically applying Fuchsian systems and isomonodromic deformations to solve the Plateau problem.

Findings

Constructed minimal disks with polygonal boundaries using Fuchsian systems.

Established the relation between boundary edge directions and monodromy.

Proved every polygon can be realized as a minimal disk boundary.

Abstract

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.