Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets

Drinovec-Drnovšek, Barbara (author), Forstnerič, Franc (author)

We show that if $E$ is a closed convex set in $\mathbb C^n$ ($n>1$) contained in a closed halfspace $H$ such that $E\cap bH$ is nonempty and bounded, then the concave domain $\Omega=\mathbb C^n\setminus E$ contains images of proper holomorphic maps $f : X \to \mathbb C^n$ from any Stein manifold $X$ of dimension $< n$, with approximation of a given map on closed compact subsets of $X$. If in addition $2 {\rm dim} X+1 \le n$ then $f$ can be chosen an embedding, and if $2 {\rm dim} X = n$, then it can be chosen an immersion. Under a stronger condition on $E$, we also obtain the interpolation property for such maps on closed complex subvarieties.

• Language: English
• Keywords: Stein manifolds, holomorphic embeddings, Oka manifold, minimal surfaces, convexity
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2023
• Number of pages: 22 str.
• Numbering: Vol. 33, iss. 6, art. 170
• PID: 20.500.12556/RUL-147581
• UDC: 517.5
• ISSN on article: 1050-6926
• DOI: 10.1007/s12220-023-01222-z
• COBISS.SI-ID: 147026947
• Publication date in RUL: 07.07.2023

Record is a part of a journal

• Title: The journal of geometric analysis
• Shortened title: J. geom. anal.
• Publisher: Springer Nature, Mathematica Josephina, Inc.
• ISSN: 1050-6926
• COBISS.SI-ID: 30685696

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: Steinove mnogoterosti, holomorfne vložitve, Oka mnogoterosti, minimalne ploskve, konveksnost