A random cover of a compact hyperbolic surface has relative spectral gap $$\frac{3}{16}-\varepsilon $$ 3 16 - ε

IF 2.4 1区 数学 Q1 MATHEMATICS Geometric and Functional Analysis Pub Date : 2022-05-17 DOI:10.1007/s00039-022-00602-x
Michael Magee, Frédéric Naud, Doron Puder
{"title":"A random cover of a compact hyperbolic surface has relative spectral gap $$\\frac{3}{16}-\\varepsilon $$ 3 16 - ε","authors":"Michael Magee, Frédéric Naud, Doron Puder","doi":"10.1007/s00039-022-00602-x","DOIUrl":null,"url":null,"abstract":"<p>Let <i>X</i> be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature <span>\\(-1\\)</span>. For each <span>\\(n\\in {\\mathbf {N}}\\)</span>, let <span>\\(X_{n}\\)</span> be a random degree-<i>n</i> cover of <i>X</i> sampled uniformly from all degree-<i>n</i> Riemannian covering spaces of <i>X</i>. An eigenvalue of <i>X</i> or <span>\\(X_{n}\\)</span> is an eigenvalue of the associated Laplacian operator <span>\\(\\Delta _{X}\\)</span> or <span>\\(\\Delta _{X_{n}}\\)</span>. We say that an eigenvalue of <span>\\(X_{n}\\)</span> is <i>new </i>if it occurs with greater multiplicity than in <i>X</i>. We prove that for any <span>\\(\\varepsilon &gt;0\\)</span>, with probability tending to 1 as <span>\\(n\\rightarrow \\infty \\)</span>, there are no new eigenvalues of <span>\\(X_{n}\\)</span> below <span>\\(\\frac{3}{16}-\\varepsilon \\)</span>. We conjecture that the same result holds with <span>\\(\\frac{3}{16}\\)</span> replaced by <span>\\(\\frac{1}{4}\\)</span>.\n</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"18 1","pages":""},"PeriodicalIF":2.4000,"publicationDate":"2022-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"10","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Geometric and Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00039-022-00602-x","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 10

Abstract

Let X be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature \(-1\). For each \(n\in {\mathbf {N}}\), let \(X_{n}\) be a random degree-n cover of X sampled uniformly from all degree-n Riemannian covering spaces of X. An eigenvalue of X or \(X_{n}\) is an eigenvalue of the associated Laplacian operator \(\Delta _{X}\) or \(\Delta _{X_{n}}\). We say that an eigenvalue of \(X_{n}\) is new if it occurs with greater multiplicity than in X. We prove that for any \(\varepsilon >0\), with probability tending to 1 as \(n\rightarrow \infty \), there are no new eigenvalues of \(X_{n}\) below \(\frac{3}{16}-\varepsilon \). We conjecture that the same result holds with \(\frac{3}{16}\) replaced by \(\frac{1}{4}\).

Abstract Image

查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
紧致双曲曲面的随机覆盖层具有相对谱隙$$\frac{3}{16}-\varepsilon $$ 3 16 - ε
设X为紧连双曲曲面,即具有常曲率黎曼度规\(-1\)的紧连可定向光滑曲面。对于每个\(n\in {\mathbf {N}}\),设\(X_{n}\)是X的随机n次覆盖,从X的所有n次黎曼覆盖空间中均匀抽样。X或\(X_{n}\)的特征值是相关拉普拉斯算子\(\Delta _{X}\)或\(\Delta _{X_{n}}\)的特征值。如果一个特征值\(X_{n}\)出现的多重性大于x,我们就说它是新的。我们证明对于任何\(\varepsilon >0\),当概率趋向于1为\(n\rightarrow \infty \)时,在\(\frac{3}{16}-\varepsilon \)以下不存在新的特征值\(X_{n}\)。我们推测,用\(\frac{1}{4}\)代替\(\frac{3}{16}\)也会得到同样的结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
CiteScore
3.70
自引率
4.50%
发文量
34
审稿时长
6-12 weeks
期刊介绍: Geometric And Functional Analysis (GAFA) publishes original research papers of the highest quality on a broad range of mathematical topics related to geometry and analysis. GAFA scored in Scopus as best journal in "Geometry and Topology" since 2014 and as best journal in "Analysis" since 2016. Publishes major results on topics in geometry and analysis. Features papers which make connections between relevant fields and their applications to other areas.
期刊最新文献
A Continuous Cusp Closing Process for Negative Kähler-Einstein Metrics On the Shapes of Rational Lemniscates Uniqueness of Tangent Flows at Infinity for Finite-Entropy Shortening Curves On the Spielman-Teng Conjecture Suppression of Chemotactic Singularity by Buoyancy
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1