Non-compact Riemann surfaces are equilaterally triangulable

• 成果类型: Article
• 署名作者: Bishop, Christopher J.; Rempe, Lasse
• 署名单位: State University of New York (SUNY) System; Stony Brook University; University of Manchester
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910; 1432-1297
• DOI: 10.1007/s00222-025-01375-4
• 发表日期: 2026-04
• 页码: 1-43
• 关键词: wandering domains; AREA DISTORTION; geodesics

摘要

We show that every open Riemann surface X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$X$\end{document} can be obtained by glueing together a countable collection of equilateral triangles, in such a way that every vertex belongs to finitely many triangles. Equivalently, X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$X$\end{document} is a Belyi surface: There exists a holomorphic branched covering f:X -> C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f\colon X\to \hat{\mathbb{C}}$\end{document} that is branched only over -1, 1 and infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\infty $\end{document}. It follows that every Riemann surface is a branched cover of the sphere, branched only over finitely many points.