This website requires JavaScript.

Similarity surfaces, connections, and the measurable Riemann mapping theorem

Arnaud Ch\'eritatGuillaume Tahar
Dec 2022
摘要
This article studies a particular process that approximates solutions of theBeltrami equation (straightening of ellipse fields, a.k.a. measurable Riemannmapping theorem) on $\mathbb{C}$. It passes through the introduction of asequence of similarity surfaces constructed by gluing polygons, and we explainthe relation between their conformal uniformization and the Schwarz-Christoffelformula. Numerical aspects, in particular the efficiency of the process, arenot studied, but we draw interesting theoretical consequences. First, we givean independent proof of the analytic dependence, on the data (the Beltramiform), of the solution of the Beltrami equation (Ahlfors-Bers theorem). Forthis we prove, without using the Ahlfors-Bers theorem, the holomorphicdependence, with respect to the polygons, of the Christoffel symbol appearingin the Schwarz-Christoffel formula. Second, these Christoffel symbols define asequence 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 paralleltransport associated to a particular affine connection, which we identify.
展开全部
图表提取

暂无人提供速读十问回答

论文十问由沈向洋博士提出,鼓励大家带着这十个问题去阅读论文,用有用的信息构建认知模型。写出自己的十问回答,还有机会在当前页面展示哦。

Q1论文试图解决什么问题?
Q2这是否是一个新的问题?
Q3这篇文章要验证一个什么科学假设?
0
被引用
笔记
问答