arXiv subjects
Publications and source records attributed to Łukasz Kosiński.
Studying the behavior of real and complex geodesics we provide sharp estimates for the Kobayashi distance, the Lempert function, and the Carathéodory distance on $\mathcal{C}^{2,α}$-smooth strongly pseudoconvex domains. Similar estimates are also provided for the Bergman distance on strongly pseudoconvex domains with $\mathcal{C}^{3,1}$-boundary.
Consider a holomorphic map $F: D \to G$ between two domains in ${\mathbb C}^N$. Let $\mathcal F$ denote a family of geodesics for the Kobayashi distance, such that $F$ acts as an isometry on each element of $\mathcal F$. This paper is dedicated to characterizing the scenarios in which the aforementioned condition implies that $F$ is a biholomorphism. Specifically, we establish this when $D$ is a complete hyperbolic domain, and $\mathcal F$ comprises all geodesic segments originating from a specific point. Another case is when $D$ and $G$ are $C^{2+α}$-smooth bounded pseudoconvex domains, and $\mathcal F$ consists of all geodesic rays converging at a designated boundary point of $D$. Furthermore, we provide examples to demonstrate that these assumptions are essentially optimal.
We prove that in a strongly pseudoconvex domain with smooth boundary, then the length of a geodesic for the Kobayashi-Royden infinitesimal metric between two points is bounded by a constant multiple of the Euclidean distance between the points.
It is proved for a strongly pseudoconvex domain $D$ in $\Bbb C^d$ with $\mathcal C^{2,α}$-smooth boundary that any complex geodesic through every two close points of $D$ sufficiently close to $\partial D$ and whose difference is non-tangential to $\partial D$ intersect a compact subset of $D$ that depends only on the rate of non-tangentiality. As an application, a lower bound for the Kobayashi distance is obtained.
We characterize polynomials that are cyclic in Dirichlet-type spaces in the unit ball in $\mathbb C^2$
In this paper we study the following "slice rigidity property": given two Kobayashi complete hyperbolic manifolds $M, N$ and a collection of complex geodesics $\mathcal F$ of $M$, when is it true that every holomorphic map $F:M\to N$ which maps isometrically every complex geodesic of $\mathcal F$ onto a complex geodesic of $N$ is a biholomorphism? Among other things, we prove that this is the case if $M, N$ are smooth bounded strictly (linearly) convex domains, every element of $\mathcal F$ contains a given point of $\overline{M}$ and $\mathcal F$ spans all of $M$. More general results are provided in dimension $2$ and for the unit ball.
Given a domain of holomorphy $D$ in $\mathbb{C}^N$, $N\geq 2$, we show that the set of holomorphic functions in $D$ whose cluster sets along any finite length paths to the boundary of $D$ is maximal, is residual, densely lineable and spaceable in the space $\mathcal{O}(D)$ of holomorphic functions in $D$. Besides, if $D$ is a strictly pseudoconvex domain in $\mathbb{C}^N$, and if a suitable family of smooth curves $γ(x,r)$, $x\in bD$, $r\in [0,1)$, ending at a point of $bD$ is given, then we exhibit a spaceable, densely lineable and residual subset of $\mathcal{O}(D)$, every element $f$ of which satisfies the following property: For any measurable function $h$ on $bD$, there exists a sequence $(r_n)_n \in [0,1)$ tending to $1$, such that \[ f\circ γ(x,r_n) \rightarrow h (x),\,n\rightarrow \infty, \] for almost every $x$ in $bD$.
We build in a given pseudoconvex (Runge) domain $D$ of $\mathbb{C}^N$ a $\mathcal O(D)$ convex set $Γ$, every connected component of which is a holomorphically contractible (convex) compact set, enjoying the property that any continuous path $γ:[0,1)\rightarrow D$ with $\lim _{r\rightarrow 1}γ(r)\in \partial D$ and omitting $Γ$ has infinite length. This solves a problem left open in a recent paper by Alarcón and Forstnerič.
For families of continuous plurisubharmonic functions we show that, in a local sense, separately bounded above implies bounded above.
We solve a three point Nevanlinna-Pick problem in the Euclidean ball. In particular, we determine a class of rational functions that interpolate this problem.
In the paper we discuss three different notions of extremal holomorphic mappings: weak $m$-extremals, $m$-extremals and $m$-complex geodesics. We discuss relations between them in general case and in the special cases of unit ball, classical Cartan domains, symmetrised bidisc and tetrablock. In particular we show that weak $3$-extremal maps in the symmetrised bidisc are rational thus giving the (partial) answer to a problem posed in a recent paper by J. Agler, Z. Lykova and N. J. Young.
In the paper we study the geometry of semitube domains in $\mathbb C^2$. In particular, we extend the result of Burgués and Dwilewicz for semitube domains dropping out the smoothness assumption. We also prove various properties of non-smooth pseudoconvex semitube domains obtaining among others a relation between pseudoconvexity of a semitube domain and the number of connected components of its vertical slices. Finally, we present an example showing that there is a non-convex domain in $\mathbb C^n$ such that its image under arbitrary isometry is pseudoconvex.
We give a complete characterization of polynomials in two complex variables that are cyclic with respect to the coordinate shifts acting on Dirichlet-type spaces in the bidisk, which include the Hardy space and the Dirichlet space of the bidisk. The cyclicity of a polynomial depends on both the size and nature of the zero set of the polynomial on the distinguished boundary. The techniques in the proof come from real analytic function theory, determinantal representations for stable polynomials, and harmonic analysis on curves