{"title":"$\\mathbb{R}^4$ 中与超平面对称的封闭拉格朗日自收缩器","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":"45 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":\"45 1\",\"pages\":\"\"},\"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}","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}
Closed Lagrangian self-shrinkers in $\mathbb{R}^4$ symmetric with respect to a hyperplane
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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