{"title":"度量曲面上的有限变形映射","authors":"Damaris Meier, Kai Rajala","doi":"10.1007/s00208-024-02972-z","DOIUrl":null,"url":null,"abstract":"<p>We investigate basic properties of <i>mappings of finite distortion</i> <span>\\(f:X \\rightarrow \\mathbb {R}^2\\)</span>, where <i>X</i> is any <i>metric surface</i>, i.e., metric space homeomorphic to a planar domain with locally finite 2-dimensional Hausdorff measure. We introduce <i>lower gradients</i>, which complement the upper gradients of Heinonen and Koskela, to study the distortion of non-homeomorphic maps on metric spaces. We extend the Iwaniec-Šverák theorem to metric surfaces: a non-constant <span>\\(f:X \\rightarrow \\mathbb {R}^2\\)</span> with locally square integrable upper gradient and locally integrable distortion is continuous, open and discrete. We also extend the Hencl-Koskela theorem by showing that if <i>f</i> is moreover injective then <span>\\(f^{-1}\\)</span> is a Sobolev map.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"9 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Mappings of finite distortion on metric surfaces\",\"authors\":\"Damaris Meier, Kai Rajala\",\"doi\":\"10.1007/s00208-024-02972-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We investigate basic properties of <i>mappings of finite distortion</i> <span>\\\\(f:X \\\\rightarrow \\\\mathbb {R}^2\\\\)</span>, where <i>X</i> is any <i>metric surface</i>, i.e., metric space homeomorphic to a planar domain with locally finite 2-dimensional Hausdorff measure. We introduce <i>lower gradients</i>, which complement the upper gradients of Heinonen and Koskela, to study the distortion of non-homeomorphic maps on metric spaces. We extend the Iwaniec-Šverák theorem to metric surfaces: a non-constant <span>\\\\(f:X \\\\rightarrow \\\\mathbb {R}^2\\\\)</span> with locally square integrable upper gradient and locally integrable distortion is continuous, open and discrete. We also extend the Hencl-Koskela theorem by showing that if <i>f</i> is moreover injective then <span>\\\\(f^{-1}\\\\)</span> is a Sobolev map.</p>\",\"PeriodicalId\":18304,\"journal\":{\"name\":\"Mathematische Annalen\",\"volume\":\"9 1\",\"pages\":\"\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-09-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematische Annalen\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00208-024-02972-z\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematische Annalen","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00208-024-02972-z","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
We investigate basic properties of mappings of finite distortion\(f:X \rightarrow \mathbb {R}^2\), where X is any metric surface, i.e., metric space homeomorphic to a planar domain with locally finite 2-dimensional Hausdorff measure. We introduce lower gradients, which complement the upper gradients of Heinonen and Koskela, to study the distortion of non-homeomorphic maps on metric spaces. We extend the Iwaniec-Šverák theorem to metric surfaces: a non-constant \(f:X \rightarrow \mathbb {R}^2\) with locally square integrable upper gradient and locally integrable distortion is continuous, open and discrete. We also extend the Hencl-Koskela theorem by showing that if f is moreover injective then \(f^{-1}\) is a Sobolev map.
期刊介绍:
Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück und Nigel Hitchin.
The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin.
Since 1868 the name Mathematische Annalen stands for a long tradition and high quality in the publication of mathematical research articles. Mathematische Annalen is designed not as a specialized journal but covers a wide spectrum of modern mathematics.