论文标题
相似性表面，连接和可测量的Riemann映射定理
Similarity surfaces, connections, and the measurable Riemann mapping theorem
论文作者
论文摘要
本文研究了一个特定的过程，该过程近似于$ \ mathbb {c} $上近似Beltrami方程的解决方案（椭圆形磁场的拉直，又称可测量的Riemann映射定理）。它通过粘合多边形构建的一系列相似性表面的序列，我们解释了它们的保形均匀化与Schwarz-Christoffel公式之间的关系。数值方面，尤其是过程的效率，没有研究，但我们会提出有趣的理论后果。首先，我们对Beltrami方程（AHLFORS-BERS定理）解决方案的数据（Beltrami形式）的分析依赖性提供了独立的证明。为此，我们证明，不使用Ahlfors-bers定理，关于多边形的全态依赖性，与Schwarz-Christoffel公式中出现的ChristOffel符号相对于多边形。其次，这些克里斯托佛尔符号定义了一系列在范围内的平行传输序列，在具有$ c^2 $带有紧凑型支持的数据的情况下，我们证明它会收敛到与特定仿射连接相关的平行传输，我们可以识别出来。
This article studies a particular process that approximates solutions of the Beltrami equation (straightening of ellipse fields, a.k.a. measurable Riemann mapping theorem) on $\mathbb{C}$. It passes through the introduction of a sequence of similarity surfaces constructed by gluing polygons, and we explain the relation between their conformal uniformization and the Schwarz-Christoffel formula. Numerical aspects, in particular the efficiency of the process, are not studied, but we draw interesting theoretical consequences. First, we give an independent proof of the analytic dependence, on the data (the Beltrami form), of the solution of the Beltrami equation (Ahlfors-Bers theorem). For this we prove, without using the Ahlfors-Bers theorem, the holomorphic dependence, with respect to the polygons, of the Christoffel symbol appearing in the Schwarz-Christoffel formula. Second, these Christoffel symbols define a sequence of parallel transports on the range, and in the case of a data that is $C^2$ with compact support, we prove that it converges to the parallel transport associated to a particular affine connection, which we identify.
