{"title":"The periodic Plateau problem and its application","authors":"Jaigyoung Choe","doi":"10.1007/s10455-025-09993-0","DOIUrl":null,"url":null,"abstract":"<div><p>Given a noncompact disconnected periodic curve <span>\\(\\Gamma \\)</span> of infinite length with two components and no self-intersection in <span>\\(\\mathbb R^3\\)</span>, it is proved that there exists a noncompact simply connected periodic minimal surface spanning <span>\\(\\Gamma \\)</span>. As an application, it is shown that for any tetrahedron <i>T</i> with dihedral angles <span>\\(\\le 90^\\circ \\)</span>, there exist four embedded minimal annuli in <i>T</i>, which are perpendicular to <span>\\(\\partial T\\)</span> along their boundary. It is also proved that every Platonic solid of <span>\\(\\mathbb R^3\\)</span> contains a free boundary embedded minimal surface of genus zero.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":"67 3","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2025-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Global Analysis and Geometry","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10455-025-09993-0","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Given a noncompact disconnected periodic curve \(\Gamma \) of infinite length with two components and no self-intersection in \(\mathbb R^3\), it is proved that there exists a noncompact simply connected periodic minimal surface spanning \(\Gamma \). As an application, it is shown that for any tetrahedron T with dihedral angles \(\le 90^\circ \), there exist four embedded minimal annuli in T, which are perpendicular to \(\partial T\) along their boundary. It is also proved that every Platonic solid of \(\mathbb R^3\) contains a free boundary embedded minimal surface of genus zero.
期刊介绍：
This journal examines global problems of geometry and analysis as well as the interactions between these fields and their application to problems of theoretical physics. It contributes to an enlargement of the international exchange of research results in the field.
The areas covered in Annals of Global Analysis and Geometry include: global analysis, differential geometry, complex manifolds and related results from complex analysis and algebraic geometry, Lie groups, Lie transformation groups and harmonic analysis, variational calculus, applications of differential geometry and global analysis to problems of theoretical physics.
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