Plateau问题、Fuchsiennes方程和Riemann-Hilbert问题
Problème de Plateau, équations fuchsiennes et problème de Riemann-Hilbert
摘要：
本文研究三维欧几里得空间中多边形边界曲线的Plateau问题。它依赖于勒内·卡尼尔开发的方法，并于1928年发表在一篇今天似乎完全被遗忘的论文中。Garnier的方法比变分方法更具有几何性和构造性，并且它提供了没有分支点的最小圆盘。然而，它有时真的很复杂，甚至模糊或不完整。遵循卡尼尔最初的想法，我们提出了一个新的证明，该证明不仅要完整，而且比他的证明更简单、更现代。这项工作主要依赖于Fuchsian系统的系统使用，以及我们在这些系统的现实和它们的单值性之间建立的关系。Garnier的方法是基于以下事实：使用极小曲面的旋量Weierstrass表示，我们可以将定义在黎曼球面上的实Fuchsian二阶方程与具有多边形边界曲线的每个极小圆盘相关联。方程的单值性由边界边的定向方向决定。为了解决高原问题，我们因此被引导来解决一个黎曼-希尔伯特问题。然后，我们进行了两个步骤：首先，通过isomonodromic变形，我们构造和描述的家庭的所有最小的磁盘与多边形边界曲线的给定方向。然后利用这一描述研究了它们的边界曲线的边长，证明了每一个多边形都是极小圆盘的边界。
英文摘要：
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 René 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
