{"title":"The parametric Willmore flow","authors":"Francesco Palmurella, Tristan Rivière","doi":"10.1515/crelle-2024-0011","DOIUrl":null,"url":null,"abstract":"\n <jats:p>We establish a minimal positive existence time of the parametric Willmore flow for any smooth initial data (smooth immersion of a closed oriented surface).\nThe minimal existence time is a function exclusively of geometric data which in particular are all well defined for general weak Lipschitz <jats:inline-formula>\n <jats:alternatives>\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:msup>\n <m:mi>W</m:mi>\n <m:mrow>\n <m:mn>2</m:mn>\n <m:mo>,</m:mo>\n <m:mn>2</m:mn>\n </m:mrow>\n </m:msup>\n </m:math>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0011_ineq_0001.png\" />\n <jats:tex-math>W^{2,2}</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula> immersions.\nThis fact opens in particular the possibility for defining the Willmore flow for weak Lipschitz <jats:inline-formula>\n <jats:alternatives>\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:msup>\n <m:mi>W</m:mi>\n <m:mrow>\n <m:mn>2</m:mn>\n <m:mo>,</m:mo>\n <m:mn>2</m:mn>\n </m:mrow>\n </m:msup>\n </m:math>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0011_ineq_0001.png\" />\n <jats:tex-math>W^{2,2}</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula> initial data.</jats:p>","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":"128 4","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal für die reine und angewandte Mathematik (Crelles Journal)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/crelle-2024-0011","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
We establish a minimal positive existence time of the parametric Willmore flow for any smooth initial data (smooth immersion of a closed oriented surface).
The minimal existence time is a function exclusively of geometric data which in particular are all well defined for general weak Lipschitz W2,2W^{2,2} immersions.
This fact opens in particular the possibility for defining the Willmore flow for weak Lipschitz W2,2W^{2,2} initial data.
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