{"title":"具有康托尔端点的双曲空间和德西特空间中的 $$text {CMC-1}$$ 曲面","authors":"Ildefonso Castro-Infantes, Jorge Hidalgo","doi":"10.1007/s00009-024-02707-z","DOIUrl":null,"url":null,"abstract":"<p>We prove that on every compact Riemann surface <i>M</i>, there is a Cantor set <span>\\(C \\subset M\\)</span> such that <span>\\(M{ \\setminus }C\\)</span> admits a proper conformal constant mean curvature one (<span>\\(\\text {CMC-1}\\)</span>) immersion into hyperbolic 3-space <span>\\(\\mathbb {H}^3\\)</span>. Moreover, we obtain that every bordered Riemann surface admits an almost proper <span>\\(\\text {CMC-1}\\)</span> face into de Sitter 3-space <span>\\(\\mathbb {S}_1^3\\)</span>, and we show that on every compact Riemann surface <i>M</i>, there is a Cantor set <span>\\(C \\subset M\\)</span> such that <span>\\(M {\\setminus } C\\)</span> admits an almost proper <span>\\(\\text {CMC-1}\\)</span> face into <span>\\(\\mathbb {S}_1^3\\)</span>. These results follow from different uniform approximation theorems for holomorphic null curves in <span>\\(\\mathbb {C}^2 \\times \\mathbb {C}^*\\)</span> that we also establish in this paper.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"$$\\\\text {CMC-1}$$ Surfaces in Hyperbolic and de Sitter Spaces with Cantor Ends\",\"authors\":\"Ildefonso Castro-Infantes, Jorge Hidalgo\",\"doi\":\"10.1007/s00009-024-02707-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We prove that on every compact Riemann surface <i>M</i>, there is a Cantor set <span>\\\\(C \\\\subset M\\\\)</span> such that <span>\\\\(M{ \\\\setminus }C\\\\)</span> admits a proper conformal constant mean curvature one (<span>\\\\(\\\\text {CMC-1}\\\\)</span>) immersion into hyperbolic 3-space <span>\\\\(\\\\mathbb {H}^3\\\\)</span>. Moreover, we obtain that every bordered Riemann surface admits an almost proper <span>\\\\(\\\\text {CMC-1}\\\\)</span> face into de Sitter 3-space <span>\\\\(\\\\mathbb {S}_1^3\\\\)</span>, and we show that on every compact Riemann surface <i>M</i>, there is a Cantor set <span>\\\\(C \\\\subset M\\\\)</span> such that <span>\\\\(M {\\\\setminus } C\\\\)</span> admits an almost proper <span>\\\\(\\\\text {CMC-1}\\\\)</span> face into <span>\\\\(\\\\mathbb {S}_1^3\\\\)</span>. These results follow from different uniform approximation theorems for holomorphic null curves in <span>\\\\(\\\\mathbb {C}^2 \\\\times \\\\mathbb {C}^*\\\\)</span> that we also establish in this paper.</p>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-08-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00009-024-02707-z\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00009-024-02707-z","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
$$\text {CMC-1}$$ Surfaces in Hyperbolic and de Sitter Spaces with Cantor Ends
We prove that on every compact Riemann surface M, there is a Cantor set \(C \subset M\) such that \(M{ \setminus }C\) admits a proper conformal constant mean curvature one (\(\text {CMC-1}\)) immersion into hyperbolic 3-space \(\mathbb {H}^3\). Moreover, we obtain that every bordered Riemann surface admits an almost proper \(\text {CMC-1}\) face into de Sitter 3-space \(\mathbb {S}_1^3\), and we show that on every compact Riemann surface M, there is a Cantor set \(C \subset M\) such that \(M {\setminus } C\) admits an almost proper \(\text {CMC-1}\) face into \(\mathbb {S}_1^3\). These results follow from different uniform approximation theorems for holomorphic null curves in \(\mathbb {C}^2 \times \mathbb {C}^*\) that we also establish in this paper.
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