On Cheeger constants of hyperbolic surfaces

• 成果类型: Article
• 署名作者: Budzinski, Thomas; Curien, Nicolas; Petri, Bram
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Ecole Normale Superieure de Lyon (ENS de LYON); Universite Paris Saclay; Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); Universite Paris Cite; Sorbonne Universite; Institut Universitaire de France
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01361-w
• 发表日期: 2025
• 页码: 511-530
• 关键词: riemann; EIGENVALUE

摘要

It is a well-known result due to Bollob & aacute;s that the maximal Cheeger constant of large d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d$\end{document}-regular graphs cannot be close to the Cheeger constant of the d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d$\end{document}-regular tree. We prove analogously that the Cheeger constant of closed hyperbolic surfaces of large genus is bounded from above by 2/pi approximate to 0.63\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2/\pi \approx 0.63$\end{document}... which is strictly less than the Cheeger constant of the hyperbolic plane. The proof uses a random construction based on a Poisson-Voronoi tessellation of the surface with a vanishing intensity.