{"title":"The Loewner–Kufarev energy and foliations by Weil–Petersson quasicircles","authors":"Fredrik Viklund, Yilin Wang","doi":"10.1112/plms.12582","DOIUrl":null,"url":null,"abstract":"We study foliations by chord–arc Jordan curves of the twice punctured Riemann sphere <mjx-container aria-label=\"double struck upper C minus StartSet 0 EndSet\" ctxtmenu_counter=\"0\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,5\" data-semantic-content=\"1\" data-semantic- data-semantic-role=\"subtraction\" data-semantic-speech=\"double struck upper C minus StartSet 0 EndSet\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-font=\"double-struck\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"numbersetletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"infixop,∖\" data-semantic-parent=\"6\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" rspace=\"4\" space=\"4\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"3\" data-semantic-content=\"2,4\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"set singleton\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"5\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"5\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/18b294c8-16a0-4c92-85bd-1f96c88b00a8/plms12582-math-0001.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,5\" data-semantic-content=\"1\" data-semantic-role=\"subtraction\" data-semantic-speech=\"double struck upper C minus StartSet 0 EndSet\" data-semantic-type=\"infixop\"><mi data-semantic-=\"\" data-semantic-font=\"double-struck\" data-semantic-parent=\"6\" data-semantic-role=\"numbersetletter\" data-semantic-type=\"identifier\" mathvariant=\"double-struck\">C</mi><mo data-semantic-=\"\" data-semantic-operator=\"infixop,∖\" data-semantic-parent=\"6\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\">∖</mo><mrow data-semantic-=\"\" data-semantic-children=\"3\" data-semantic-content=\"2,4\" data-semantic-parent=\"6\" data-semantic-role=\"set singleton\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"5\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">{</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\">0</mn><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"5\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">}</mo></mrow></mrow>$\\mathbb {C} \\setminus \\lbrace 0\\rbrace$</annotation></semantics></math></mjx-assistive-mml></mjx-container> using the Loewner–Kufarev equation. We associate to such a foliation a function on the plane that describes the “local winding” along each leaf. Our main theorem is that this function has finite Dirichlet energy if and only if the Loewner driving measure <mjx-container aria-label=\"rho\" ctxtmenu_counter=\"1\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"greekletter\" data-semantic-speech=\"rho\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/bb24b5d9-08f0-438c-998d-3ddf73cde132/plms12582-math-0002.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"greekletter\" data-semantic-speech=\"rho\" data-semantic-type=\"identifier\">ρ</mi>$\\rho$</annotation></semantics></math></mjx-assistive-mml></mjx-container> has finite Loewner–Kufarev energy, defined by","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"55 1","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1112/plms.12582","RegionNum":1,"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 foliations by chord–arc Jordan curves of the twice punctured Riemann sphere using the Loewner–Kufarev equation. We associate to such a foliation a function on the plane that describes the “local winding” along each leaf. Our main theorem is that this function has finite Dirichlet energy if and only if the Loewner driving measure has finite Loewner–Kufarev energy, defined by
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