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Query: "keywords" (Stein manifold) .

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1.
Proper holomorphic embeddings with small limit sets
Franc Forstnerič, 2024, original scientific article

Abstract: Let $X$ be a Stein manifold of dimension $n\ge 1$. Given a continuous positive increasing function $h$ on ${\mathbb R}_+ = [0,\infty)$ with $\lim_{t\to\infty} h(t)=\infty$, we construct a proper holomorphic embedding $f=(z,w):X \hookrightarrow {\mathbb C}^{n+1}\times {\mathbb C}^n$ satisfying $|w(x)|Keywords: Stein manifold, proper holomorphic embedding
Published in DiRROS: 13.05.2024; Views: 36; Downloads: 14
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2.
Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
Barbara Drinovec-Drnovšek, Franc Forstnerič, 2023, original scientific article

Abstract: 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 holomorphicmaps $f : X \to \mathbb C^n$ from any Stein manifold $X$ of dimension $< n$, with approximation of a givenmap 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.
Keywords: Stein manifolds, holomorphic embeddings, Oka manifold, minimal surfaces, convexity
Published in DiRROS: 15.03.2024; Views: 113; Downloads: 55
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3.
Recent developments on Oka manifolds
Franc Forstnerič, 2023, review article

Abstract: In this paper we present the main developments in Oka theory since the publication of my book "Stein Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex Analysis)", 2nd ed., Springer, 2017. We also give several new results, examples and constructions of Oka domains in Euclidean and projective spaces. Furthermore, we show that for $n > 1$ the fibre $\mathbb C^n$ in a Stein family can degenerate to a non-Oka fibre, thereby answering a question of Takeo Ohsawa. Several open problems are discussed.
Keywords: Oka manifold, Oka map, Stein manifold, elliptic manifold, algebraically subelliptic manifold, Calabi–Yau manifold, density property
Published in DiRROS: 14.03.2024; Views: 424; Downloads: 456
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