{"title":"Deformation of Kähler metrics and an eigenvalue problem for the Laplacian on a compact Kähler manifold","authors":"Kazumasa Narita","doi":"10.1007/s00229-024-01592-w","DOIUrl":null,"url":null,"abstract":"<p>We study an eigenvalue problem for the Laplacian on a compact Kähler manifold. Considering the <i>k</i>-th eigenvalue <span>\\(\\lambda _{k}\\)</span> as a functional on the space of Kähler metrics with fixed volume on a compact complex manifold, we introduce the notion of <span>\\(\\lambda _{k}\\)</span>-extremal Kähler metric. We deduce a condition for a Kähler metric to be <span>\\(\\lambda _{k}\\)</span>-extremal. As examples, we consider product Kähler manifolds, compact isotropy irreducible homogeneous Kähler manifolds and flat complex tori.</p>","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":"111 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Manuscripta Mathematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00229-024-01592-w","RegionNum":4,"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
We study an eigenvalue problem for the Laplacian on a compact Kähler manifold. Considering the k-th eigenvalue \(\lambda _{k}\) as a functional on the space of Kähler metrics with fixed volume on a compact complex manifold, we introduce the notion of \(\lambda _{k}\)-extremal Kähler metric. We deduce a condition for a Kähler metric to be \(\lambda _{k}\)-extremal. As examples, we consider product Kähler manifolds, compact isotropy irreducible homogeneous Kähler manifolds and flat complex tori.
期刊介绍:
manuscripta mathematica was founded in 1969 to provide a forum for the rapid communication of advances in mathematical research. Edited by an international board whose members represent a wide spectrum of research interests, manuscripta mathematica is now recognized as a leading source of information on the latest mathematical results.