{"title":"3-2-1 foliations for Reeb flows on the tight 3-sphere","authors":"Carolina de Oliveira","doi":"10.1090/tran/9119","DOIUrl":null,"url":null,"abstract":"<p>We study the existence of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3 minus 2 minus 1\"> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">3-2-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> foliations adapted to Reeb flows on the tight <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=\"application/x-tex\">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-sphere. These foliations admit precisely three binding orbits whose Conley-Zehnder indices are <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=\"application/x-tex\">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding=\"application/x-tex\">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1\"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding=\"application/x-tex\">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, respectively. All regular leaves are disks and annuli asymptotic to the binding orbits. Our main results provide sufficient conditions for the existence of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3 minus 2 minus 1\"> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">3-2-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> foliations with prescribed binding orbits. We also exhibit a concrete Hamiltonian on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R Superscript 4\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">\\mathbb {R}^4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> admitting <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3 minus 2 minus 1\"> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">3-2-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> foliations when restricted to suitable energy levels.</p>","PeriodicalId":23209,"journal":{"name":"Transactions of the American Mathematical Society","volume":"7 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/tran/9119","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We study the existence of 3−2−13-2-1 foliations adapted to Reeb flows on the tight 33-sphere. These foliations admit precisely three binding orbits whose Conley-Zehnder indices are 33, 22, and 11, respectively. All regular leaves are disks and annuli asymptotic to the binding orbits. Our main results provide sufficient conditions for the existence of 3−2−13-2-1 foliations with prescribed binding orbits. We also exhibit a concrete Hamiltonian on R4\mathbb {R}^4 admitting 3−2−13-2-1 foliations when restricted to suitable energy levels.
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