{"title":"Curvature varifolds with orthogonal boundary","authors":"Ernst Kuwert, Marius Müller","doi":"10.1112/jlms.12976","DOIUrl":null,"url":null,"abstract":"<p>We consider the class <span></span><math>\n <semantics>\n <mrow>\n <msubsup>\n <mi>S</mi>\n <mo>⊥</mo>\n <mi>m</mi>\n </msubsup>\n <mrow>\n <mo>(</mo>\n <mi>Ω</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>${\\bf S}^m_\\perp (\\Omega)$</annotation>\n </semantics></math> of <span></span><math>\n <semantics>\n <mi>m</mi>\n <annotation>$m$</annotation>\n </semantics></math>-dimensional surfaces in <span></span><math>\n <semantics>\n <mrow>\n <mover>\n <mi>Ω</mi>\n <mo>¯</mo>\n </mover>\n <mo>⊂</mo>\n <msup>\n <mi>R</mi>\n <mi>n</mi>\n </msup>\n </mrow>\n <annotation>$\\overline{\\Omega } \\subset {\\mathbb {R}}^n$</annotation>\n </semantics></math> that intersect <span></span><math>\n <semantics>\n <mrow>\n <mi>S</mi>\n <mo>=</mo>\n <mi>∂</mi>\n <mi>Ω</mi>\n </mrow>\n <annotation>$S = \\partial \\Omega$</annotation>\n </semantics></math> orthogonally along the boundary. A piece of an affine <span></span><math>\n <semantics>\n <mi>m</mi>\n <annotation>$m$</annotation>\n </semantics></math>-plane in <span></span><math>\n <semantics>\n <mrow>\n <msubsup>\n <mi>S</mi>\n <mo>⊥</mo>\n <mi>m</mi>\n </msubsup>\n <mrow>\n <mo>(</mo>\n <mi>Ω</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>${\\bf S}^m_\\perp (\\Omega)$</annotation>\n </semantics></math> is called an orthogonal slice. We prove estimates for the area by the <span></span><math>\n <semantics>\n <msup>\n <mi>L</mi>\n <mi>p</mi>\n </msup>\n <annotation>$L^p$</annotation>\n </semantics></math> integral of the second fundamental form in three cases: first, when <span></span><math>\n <semantics>\n <mi>Ω</mi>\n <annotation>$\\Omega$</annotation>\n </semantics></math> admits no orthogonal slices, second for <span></span><math>\n <semantics>\n <mrow>\n <mi>m</mi>\n <mo>=</mo>\n <mi>p</mi>\n <mo>=</mo>\n <mn>2</mn>\n </mrow>\n <annotation>$m = p = 2$</annotation>\n </semantics></math> if all orthogonal slices are topological disks, and finally, for all <span></span><math>\n <semantics>\n <mi>Ω</mi>\n <annotation>$\\Omega$</annotation>\n </semantics></math> if the surfaces are confined to a neighborhood of <span></span><math>\n <semantics>\n <mi>S</mi>\n <annotation>$S$</annotation>\n </semantics></math>. The orthogonality constraint has a weak formulation for curvature varifolds. We classify those varifolds of vanishing curvature. As an application, we prove for any <span></span><math>\n <semantics>\n <mi>Ω</mi>\n <annotation>$\\Omega$</annotation>\n </semantics></math> the existence of an orthogonal 2-varifold that minimizes the <span></span><math>\n <semantics>\n <msup>\n <mi>L</mi>\n <mn>2</mn>\n </msup>\n <annotation>$L^2$</annotation>\n </semantics></math> curvature in the integer rectifiable class.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12976","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12976","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
We consider the class of -dimensional surfaces in that intersect orthogonally along the boundary. A piece of an affine -plane in is called an orthogonal slice. We prove estimates for the area by the integral of the second fundamental form in three cases: first, when admits no orthogonal slices, second for if all orthogonal slices are topological disks, and finally, for all if the surfaces are confined to a neighborhood of . The orthogonality constraint has a weak formulation for curvature varifolds. We classify those varifolds of vanishing curvature. As an application, we prove for any the existence of an orthogonal 2-varifold that minimizes the curvature in the integer rectifiable class.
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
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