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Oka domains in Euclidean spacesFranc Forstnerič,
Erlend Fornæss Wold, 2024, original scientific article
Abstract: 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.
Keywords: Oka manifold, hyperbolic manifolds, density property, projectively convex sets
Published in DiRROS: 19.02.2024; Views: 219; Downloads: 74
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