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2. Proper holomorphic maps in Euclidean spaces avoiding unbounded convex setsBarbara 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: 117; Downloads: 57
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3. 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: 232; Downloads: 76
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4. Complete nonsingular holomorphic foliations on Stein manifoldsAntonio Alarcón, Franc Forstnerič, 2024, original scientific article Abstract: 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.
Keywords: Stein manifolds, complete holomorphic foliations, density property
Published in DiRROS: 19.02.2024; Views: 190; Downloads: 68
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5. Fractional Sobolev spaces with kernel function on compact Riemannian manifoldsAhmed Aberqi, Abdesslam Ouaziz, Dušan Repovš, 2024, original scientific article Abstract: In this paper, a new class of Sobolev spaces with kernel function satisfying a Lévy-integrability-type condition on compact Riemannian manifolds is presented. We establish the properties of separability, reflexivity, and completeness. An embedding result is also proved. As an application, we prove the existence of solutions for a nonlocal elliptic problem involving the fractional $p(\cdot, \cdot)$-Laplacian operator. As one of the main tools, topological degree theory is applied.
Keywords: nonlinear elliptic problem, fractional Sobolev spaces, kernel function, Lévy-integrability condition, compact Riemannian manifolds, existence of solutions, topological degree theory
Published in DiRROS: 19.02.2024; Views: 164; Downloads: 64
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6. Thermal fatigue degradation progress in SiMo ductile cast iron under oxidation conditionsMilan Terčelj, Jaka Burja, Goran Kugler, Primož Mrvar, 2023, original scientific article Keywords: metallurgical engineering, exhaust manifolds, cast irons, casting, microscopic characterization and microanalysis, thermal fatigue, material defect, microstructural control
Published in DiRROS: 07.02.2024; Views: 222; Downloads: 157
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