{"title":"圆柱体中螺旋面的新特征","authors":"Eunjoo Lee","doi":"10.2748/tmj.20210713","DOIUrl":null,"url":null,"abstract":"This paper characterizes a compact piece of the helicoid HC in a solid cylinder C ⊂ R from the following two perspectives. First, under reasonable conditions, HC has the smallest area among all immersed surfaces Σ with ∂Σ ⊂ d1 ∪ d2 ∪ S, where d1 and d2 are the diameters of the top and bottom disks of C and S is the side surface of C. Second, other than HC , there exists no minimal surface whose boundary consists of d1, d2, and a pair of rotationally symmetric curves γ1, γ2 on S along which it meets S orthogonally. We draw the same conclusion when the boundary curves on S are a pair of helices of a certain height.","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2021-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"New characterizations of the helicoid in a cylinder\",\"authors\":\"Eunjoo Lee\",\"doi\":\"10.2748/tmj.20210713\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This paper characterizes a compact piece of the helicoid HC in a solid cylinder C ⊂ R from the following two perspectives. First, under reasonable conditions, HC has the smallest area among all immersed surfaces Σ with ∂Σ ⊂ d1 ∪ d2 ∪ S, where d1 and d2 are the diameters of the top and bottom disks of C and S is the side surface of C. Second, other than HC , there exists no minimal surface whose boundary consists of d1, d2, and a pair of rotationally symmetric curves γ1, γ2 on S along which it meets S orthogonally. We draw the same conclusion when the boundary curves on S are a pair of helices of a certain height.\",\"PeriodicalId\":54427,\"journal\":{\"name\":\"Tohoku Mathematical Journal\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2021-08-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Tohoku Mathematical Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.2748/tmj.20210713\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Tohoku Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2748/tmj.20210713","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
New characterizations of the helicoid in a cylinder
This paper characterizes a compact piece of the helicoid HC in a solid cylinder C ⊂ R from the following two perspectives. First, under reasonable conditions, HC has the smallest area among all immersed surfaces Σ with ∂Σ ⊂ d1 ∪ d2 ∪ S, where d1 and d2 are the diameters of the top and bottom disks of C and S is the side surface of C. Second, other than HC , there exists no minimal surface whose boundary consists of d1, d2, and a pair of rotationally symmetric curves γ1, γ2 on S along which it meets S orthogonally. We draw the same conclusion when the boundary curves on S are a pair of helices of a certain height.