{"title":"Embedded complex curves in the affine plane","authors":"Antonio Alarcón, Franc Forstnerič","doi":"10.1007/s10231-023-01418-8","DOIUrl":null,"url":null,"abstract":"<div><p>This paper brings several contributions to the classical Forster–Bell–Narasimhan conjecture and the Yang problem concerning the existence of proper, almost proper, and complete injective holomorphic immersions of open Riemann surfaces in the affine plane <span>\\(\\mathbb {C}^2\\)</span> satisfying interpolation and hitting conditions. We also show that every compact Riemann surface contains a Cantor set whose complement admits a proper holomorphic embedding in <span>\\(\\mathbb {C}^2\\)</span>, and every connected domain in <span>\\(\\mathbb {C}^2\\)</span> admits complete, everywhere dense, injectively immersed complex discs. The focal point of the paper is a lemma saying for every compact bordered Riemann surface, <i>M</i>, closed discrete subset <i>E</i> of <span>\\(\\mathring{M}=M\\setminus bM\\)</span>, and compact subset <span>\\(K\\subset \\mathring{M}\\setminus E\\)</span> without holes in <span>\\(\\mathring{M}\\)</span>, any <span>\\(\\mathscr {C}^1\\)</span> embedding <span>\\(f:M\\hookrightarrow \\mathbb {C}^2\\)</span> which is holomorphic in <span>\\(\\mathring{M}\\)</span> can be approximated uniformly on <i>K</i> by holomorphic embeddings <span>\\(F:M\\hookrightarrow \\mathbb {C}^2\\)</span> which map <span>\\(E\\cup bM\\)</span> out of a given ball and satisfy some interpolation conditions.</p></div>","PeriodicalId":8265,"journal":{"name":"Annali di Matematica Pura ed Applicata","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10231-023-01418-8.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annali di Matematica Pura ed Applicata","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10231-023-01418-8","RegionNum":3,"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
This paper brings several contributions to the classical Forster–Bell–Narasimhan conjecture and the Yang problem concerning the existence of proper, almost proper, and complete injective holomorphic immersions of open Riemann surfaces in the affine plane \(\mathbb {C}^2\) satisfying interpolation and hitting conditions. We also show that every compact Riemann surface contains a Cantor set whose complement admits a proper holomorphic embedding in \(\mathbb {C}^2\), and every connected domain in \(\mathbb {C}^2\) admits complete, everywhere dense, injectively immersed complex discs. The focal point of the paper is a lemma saying for every compact bordered Riemann surface, M, closed discrete subset E of \(\mathring{M}=M\setminus bM\), and compact subset \(K\subset \mathring{M}\setminus E\) without holes in \(\mathring{M}\), any \(\mathscr {C}^1\) embedding \(f:M\hookrightarrow \mathbb {C}^2\) which is holomorphic in \(\mathring{M}\) can be approximated uniformly on K by holomorphic embeddings \(F:M\hookrightarrow \mathbb {C}^2\) which map \(E\cup bM\) out of a given ball and satisfy some interpolation conditions.
期刊介绍:
This journal, the oldest scientific periodical in Italy, was originally edited by Barnaba Tortolini and Francesco Brioschi and has appeared since 1850. Nowadays it is managed by a nonprofit organization, the Fondazione Annali di Matematica Pura ed Applicata, c.o. Dipartimento di Matematica "U. Dini", viale Morgagni 67A, 50134 Firenze, Italy, e-mail annali@math.unifi.it).
A board of Italian university professors governs the Fondazione and appoints the editors of the journal, whose responsibility it is to supervise the refereeing process. The names of governors and editors appear on the front page of each issue. Their addresses appear in the title pages of each issue.