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1.
Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
Barbara Drinovec-Drnovšek, Franc Forstnerič, 2023, izvirni znanstveni članek

Povzetek: 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.
Ključne besede: Stein manifolds, holomorphic embeddings, Oka manifold, minimal surfaces, convexity
Objavljeno v DiRROS: 15.03.2024; Ogledov: 96; Prenosov: 48
.pdf Celotno besedilo (441,34 KB)
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2.
Complete nonsingular holomorphic foliations on Stein manifolds
Antonio Alarcón, Franc Forstnerič, 2024, izvirni znanstveni članek

Povzetek: Let $X$ be a Stein manifold of complex dimension $n \ge 1$ endowed with a Riemannian metric ${\mathfrak g}$. We show that for every integer $k$ with $\left[\frac{n}{2}\right] \le k \le n-1$ there is a nonsingular holomorphic foliation of dimension $k$ on $X$ all of whose leaves are topologically closed and ${\mathfrak g}$-complete. The same is true if $1\le k \left[\frac{n}{2}\right]$ provided that there is a complex vector bundle epimorphism $TX\to X \times \mathbb{C}^{n-k}$. We also show that if $\mathcal{F}$ is a proper holomorphic foliation on $\mathbb{C}^n$ $(n > 1)$ then for any Riemannian metric ${\mathfrak g}$ on $\mathbb{C}^n$ there is a holomorphic automorphism $\Phi$ of $\mathbb{C}^n$ such that the image foliation $\Phi_*\mathcal{F}$ is ${\mathfrak g}$-complete. The analogous result is obtained on every Stein manifold with Varolin's density property.
Ključne besede: Stein manifolds, complete holomorphic foliations, density property
Objavljeno v DiRROS: 19.02.2024; Ogledov: 167; Prenosov: 59
.pdf Celotno besedilo (433,06 KB)
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