{"title":"Closed Lagrangian self-shrinkers in $\\mathbb{R}^4$ symmetric with respect to a hyperplane","authors":"Jaehoon Lee","doi":"10.4310/cag.2023.v31.n8.a3","DOIUrl":null,"url":null,"abstract":"In this paper, we prove that the closed Lagrangian self-shrinkers in $\\mathbb{R}^4$ which are symmetric with respect to a hyperplane are given by the products of Abresch–Langer curves. As a corollary, we obtain a new geometric characterization of the Clifford torus as the unique embedded closed Lagrangian self-shrinker symmetric with respect to a hyperplane in $\\mathbb{R}^4$.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Analysis and Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/cag.2023.v31.n8.a3","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
In this paper, we prove that the closed Lagrangian self-shrinkers in $\mathbb{R}^4$ which are symmetric with respect to a hyperplane are given by the products of Abresch–Langer curves. As a corollary, we obtain a new geometric characterization of the Clifford torus as the unique embedded closed Lagrangian self-shrinker symmetric with respect to a hyperplane in $\mathbb{R}^4$.
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