Related papers: Proper holomorphic maps in Euclidean spaces avoidi…

We show that the only proper-holomorphic self-maps of bounded domains in C^k whose dynamics escape to a strictly pseudoconvex point of the boundary are automorphisms of the euclidean ball. This is a Wong-Rosay type result for a sequence of…

We classify hypersurfaces of rank two of Euclidean space $\R^{n+1}$ that admit genuine isometric deformations in $\R^{n+2}$. That an isometric immersion $\hat f\colon\,M^n\to\R^{n+2}$ is a genuine isometric deformation of a hypersurface…

A complex manifold Y satisfies the Convex Approximation Property (CAP) if every holomorphic map from a neighborhood of a compact convex set K in a complex Euclidean space C^n to Y can be approximated, uniformly on K, by entire maps from C^n…

Suppose that $M$ is a Riemann surface with boundary $\partial M$, $\Lambda$ is its DN-map, and $\mathscr E:M\to\mathbb{C}^{n}$ % $\mathfrak{J}_{M}$ is a holomorphic immersion. Let $M'$ be diffeomorphic to $M$, $\partial M=\partial M'$; let…

Let D be a relatively compact strongly pseudoconvex domain in a Stein manifold, and let Y be a complex manifold. We prove that the set A(D,Y), consisting of all continuous maps from the closure of D to Y which are holomorphic in D, is a…

We prove that a closed convex subset $C$ of a complete linear metric space $X$ is polyhedral in its closed linear hull if and only if no infinite subset $A\subset X\backslash C$ can be hidden behind $C$ in the sense $[x,y]\cap C\not =…

Let $P \subset \mathbb{R}^{d}$ be a closed convex cone. Assume that $P$ is pointed, i.e. the intersection $P \cap -P=\{0\}$ and $P$ is spanning, i.e. $P-P=\mathbb{R}^{d}$. Denote the interior of $P$ by $\Omega$. Let $E$ be a product system…

In this work we prove constructively that the complement ${\mathbb R}^n\setminus{\mathcal K}$ of an $n$-dimensional unbounded convex polyhedron ${\mathcal K}\subset{\mathbb R}^n$ and the complement ${\mathbb R}^n\setminus{\rm Int}({\mathcal…

We show that there are strictly pseudoconvex, real algebraic hypersurfaces in $\bC^{n+1}$ that cannot be locally embedded into a sphere in $\bC^{N+1}$ for any $N$. In fact, we show that there are strictly pseudoconvex, real algebraic…

Let $\D$ be the unit disk. Kutzschebauch and Studer \cite{KS} recently proved that, for each continuous map $A:\overline D\to \mathrm{SL}(2,\C)$, which is holomorphic in $\D$, there exist continuous maps $E,F:\overline \D\to…

We solve the problem of simultaneously embedding properly holomorphically into $\Bbb C^2$ a whole family of $n$-connected domains $\Omega_r\subset\Bbb P^1$ such that none of the components of $\Bbb P^1\setminus\Omega_r$ reduces to a point,…

We study betweenness preserving mappings (we call them \emph{monotone}) defined on subsets of the plane. Once the domain is a convex set, such a mapping is either the restriction of a homography, or its image is contained in the union of a…

We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We prove that each Hausdorff topological space can be embedded as a closed subspace into an H-closed topological space. However,…

We prove holomorphy E sqcap C(I,varPi) to C(I,varPi) of the map (x,y) mapsto x circ [id,y] where [id,y]:I owns t mapsto (t,y(t)) for a real compact interval I, and where varPi is a complex Banach space and E is a certain locally convex…

In this article we prove first of all the nonexistence of holomorphic submersions other than covering maps between compact quotients of complex unit balls, with a proof that works equally well in a more general equivariant setting. For a…

We prove that if $f\colon\mathbb{C}^p\rightarrow\mathbb{P}^n(\mathbb{C})$ is a holomorphic mapping of maximal rank whose image lies in the Fermat hypersurface of degree $d>(n+1)\max\{n-p,1\}$, then its image is contained in a linear…

We study the approximation of maps into complex manifolds along with interpolation on certain compact subsets of the plane. Results are also obtained regarding approximation and interpolation of sections of holomorphic submersions.

We show that any compact smooth real $n$-dimensional manifold $M$ with $n\leq 11$ can be smoothly embedded into $\mathbb{C}^{n+1}$ as a polynomially convex set. In general, there is no such embedding into $\mathbb{C}^n$. This solves a…

We prove that the image of a finely holomorphic map on a fine domain in $\mathbb{C}$ is pluripolar subset of $\mathbb{C}^{n}$. We also discuss the relationship between pluripolar hulls and finely holomorphic function.

We construct holomorphic maps with a Siegel disk whose boundary is not locally connected (and is an indecomposable continuum), yet compactly contained in the domain of definition of the map. Our examples are injective and defined on a…