{"title":"最佳扁平带的存在","authors":"Simon Blatt, Matteo Raffaelli","doi":"10.1007/s12220-024-01683-w","DOIUrl":null,"url":null,"abstract":"<p>We apply the direct method of the calculus of variations to show that any nonplanar Frenet curve in <span>\\({\\mathbb {R}}^{3}\\)</span> can be extended to an infinitely narrow flat ribbon having <i>minimal</i> bending energy. We also show that, in general, minimizers are not free of planar points, yet such points must be isolated under the mild condition that the torsion does not vanish.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"58 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Existence of Optimal Flat Ribbons\",\"authors\":\"Simon Blatt, Matteo Raffaelli\",\"doi\":\"10.1007/s12220-024-01683-w\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We apply the direct method of the calculus of variations to show that any nonplanar Frenet curve in <span>\\\\({\\\\mathbb {R}}^{3}\\\\)</span> can be extended to an infinitely narrow flat ribbon having <i>minimal</i> bending energy. We also show that, in general, minimizers are not free of planar points, yet such points must be isolated under the mild condition that the torsion does not vanish.</p>\",\"PeriodicalId\":501200,\"journal\":{\"name\":\"The Journal of Geometric Analysis\",\"volume\":\"58 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-05-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"The Journal of Geometric Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s12220-024-01683-w\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Journal of Geometric Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s12220-024-01683-w","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We apply the direct method of the calculus of variations to show that any nonplanar Frenet curve in \({\mathbb {R}}^{3}\) can be extended to an infinitely narrow flat ribbon having minimal bending energy. We also show that, in general, minimizers are not free of planar points, yet such points must be isolated under the mild condition that the torsion does not vanish.
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