Oka domains in Euclidean spaces

Forstnerič, Franc (author), Wold, Erlend Fornæss (author)

In this paper, we find surprisingly small Oka domains in Euclidean spaces $\mathbb C^n$ of dimension $n>1$ at the very limit of what is possible. Under a mild geometric assumption on a closed unbounded convex set $E$ in $\mathbb C^n$, we show that $\mathbb C^n\setminus E$ is an Oka domain. In particular, there are Oka domains only slightly bigger than a halfspace, the latter being neither Oka nor hyperbolic. This gives smooth families of real hypersurfaces $\Sigma_t \subset \mathbb C^n$ for $t \in \mathbb R$ dividing $\mathbb C^n$ in an unbounded hyperbolic domain and an Oka domain such that at $t=0$, $\Sigma_0$ is a hyperplane and the character of the two sides gets reversed. More generally, we show that if $E$ is a closed set in $\mathbb C^n$ for $n>1$ whose projective closure $\overline E \subset \mathbb{CP}^n$ avoids a hyperplane $\Lambda \subset \mathbb{CP}^n$ and is polynomially convex in $\mathbb{CP}^n\setminus \Lambda\cong\mathbb C^n$, then $\mathbb C^n\setminus E$ is an Oka domain.

• Language: English
• Keywords: Oka manifold, hyperbolic manifolds, density property, projectively convex sets
• 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: 2024
• Number of pages: Str. 1801-1824
• Numbering: Vol. 2024, no. 3
• PID: 20.500.12556/RUL-154512
• UDC: 517.5
• ISSN on article: 1687-0247
• DOI: 10.1093/imrn/rnac347
• COBISS.SI-ID: 143307011
• Publication date in RUL: 19.02.2024

FORSTNERIČ, Franc and WOLD, Erlend Fornæss, 2024, Oka domains in Euclidean spaces. International mathematics research notices [online]. 2024. Vol. 2024, no. 3, p. 1801–1824. [Accessed 13 April 2025]. DOI 10.1093/imrn/rnac347. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=154512

Record is a part of a journal

• Title: International mathematics research notices
• Shortened title: Int. math. res. not.
• Publisher: Oxford University Press
• ISSN: 1687-0247
• COBISS.SI-ID: 515379993

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: mnogoterost Oka, hiperbolične mnogoterosti, lastnost gostote, projektivno konveksna množica