Embedded complex curves in the affine plane

Alarcón, Antonio (author), Forstnerič, Franc (author)

This paper brings several contributions to the classical Forster-Bell-Narasimhan conjecture and the Yang problem concerning the existence of proper and almost proper (hence complete) injective holomorphic immersions of open Riemann surfaces in the affine plane ${\mathbb C}^2$ satisfying interpolation and hitting conditions. We also show that in every compact Riemann surface there is a Cantor set whose complement admits a proper holomorphic embedding in ${\mathbb C}^2$. The focal point is a lemma saying the following. Given a compact bordered Riemann surface, $M$, a closed discrete subset $E$ of its interior ${\mathring M}=M\setminus bM$, a compact subset $K\subset {\mathring M}\setminus E$ without holes in $\mathring M$, and a ${\cal C}^1$ embedding $f: M\hookrightarrow \mathbb C^2$ which is holomorphic in $\mathring M$, we can approximate $f$ uniformly on $K$ by a holomorphic embedding $F: bM\hookrightarrow {\mathbb C}^2$ which maps $E\cup bM$ out of a given ball and satisfies some interpolation conditions.

• Language: English
• Keywords: Riemann surfaces, complex curves, complete holomorphic embedding
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication version: Version of Record
• Publication date: 01.08.2024
• Year: 2024
• Number of pages: Str. 1673-1701
• Numbering: Vol. 203, iss. 4
• PID: 20.500.12556/RUL-159621
• UDC: 517.5
• ISSN on article: 0373-3114
• DOI: 10.1007/s10231-023-01418-8
• COBISS.SI-ID: 182950147
• Publication date in RUL: 15.07.2024

Record is a part of a journal

• Title: Annali di matematica pura ed applicata
• Shortened title: Ann. mat. pura appl.
• Publisher: Springer
• ISSN: 0373-3114
• COBISS.SI-ID: 24962816

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: Riemannove ploskve, kompleksne krivulje, kompletna holomorfna vložitev