Related papers: Kobayashi-Royden vs. Hahn pseudometric in ${\Bbb C…

The Fock-Bargmann-Hartogs domain $D_{n,m}$ in $\mathbb{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-\|z\|^2},$ where $(z,w)\in \mathbb{C}^n\times \mathbb{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbb{C}^{n+m}$. This…

We present a form of Schwarz's lemma for holomorphic maps between convex domains $D_1$ and $D_2$. This result provides a lower bound on the distance between the images of relatively compact subsets of $D_1$ and the boundary of $D_2$. This…

We prove that: I. If $L$ is a $T_1$ space, $|L|>1$ and $d(L) \leq \kappa \geq \omega$, then there is a submaximal dense subspace $X$ of $L^{2^\kappa}$ such that $|X|=\Delta(X)=\kappa$; II. If $\frak{c}\leq\kappa=\kappa^\omega<\lambda$ and…

It is proved for a strongly pseudoconvex domain $D$ in $\Bbb C^d$ with $\mathcal C^{2,\alpha}$-smooth boundary that any complex geodesic through every two close points of $D$ sufficiently close to $\partial D$ and whose difference is…

In this article, we prove localization results for the Kobayashi distance of Kobayashi hyperbolic domains with local visibility property in $\mathbb{C}^d$, $d \geq 1$. This is done by proving a localization result for the Kobayashi-Royden…

We prove that for a strongly pseudoconvex domain $D\subset\mathbb C^n$, the infinitesimal Carath\'eodory metric $g_C(z,v)$ and the infinitesimal Kobayashi metric $g_K(z,v)$ coincide if $z$ is sufficiently close to $bD$ and if $v$ is…

The following result is proved: Let $D$ and $D'$ be bounded domains in $\mathbb C^n$, $\partial D$ is smooth, real-analytic, simply connected, and $\partial D'$ is connected, smooth, real-algebraic. Then there exists a proper holomorphic…

We prove that the Kobayashi distance near boundary of a pseudoconvex Reinhardt domain $D$ increases asymptotically at most like $-\log d_D+C$. Moreover, for boundary points from $\text{int}\bar{D}$ the growth does not exceed $1/2\log(-\log…

For a two dimensional bounded pseudoconvex domain of finite type, we prove uniformization theorems via K$\ddot{\operatorname{a}}$hler-Kobayashi metric or K$\ddot{\operatorname{a}}$hler-Carath$\acute{\operatorname{e}}$odory metric with…

This article considers isometries of the Kobayashi and Carath\'{e}od-ory metrics on domains in $ \mathbf{C}^n $ and the extent to which they behave like holomorphic mappings. First we prove a metric version of Poincar\'{e}'s theorem about…

Let $\Omega=\widetilde{\Omega}\setminus \overline{D}$ where $\widetilde{\Omega}$ is a bounded domain with connected complement in $\mathbb C^n$ (or more generally in a Stein manifold) and $D$ is relatively compact open subset of…

Nagano spaces are compact symmetric spaces that admit large transformation groups. They include for instance all the Grassmannians and the Einstein Universes. In this paper, we study a Kobayashi-type pseudometric on domains in real-type…

Let $D$ be a smoothly bounded pseudoconvex domain in $\mathbf C^n$, $n > 1$. Using the Robin function $\La(p)$ that arises from the Green function $G(z, p)$ for $D$ with pole at $p \in D$ associated with the standard sum-of-squares…

We provide counterexamples to the stable equivalence problem in every dimension $d\geq2$. That means that we construct hypersurfaces $H_1, H_2\subset\mathbb{C}^{d+1}$ whose cylinders $H_1\times\mathbb{C}$ and $H_2\times\mathbb{C}$ are…

This paper deals with the reducibility property of semidirect products of the form $\bf V*\bf D$ relatively to graph equation systems, where $\bf D$ denotes the pseudovariety of definite semigroups. We show that, if the pseudovariety $\bf…

We study proper holomorphic maps between bounded symmetric domains $D$ and $\Omega$. In particular, when $D$ and $\Omega$ are of the same rank $\ge 2$ such that all irreducible factors of $D$ are of rank $\ge 2$, we prove that any proper…

The purpose of this article is to consider two themes both of which emanate from and involve the Kobayashi and the Carath\'eodory metric. First we study the biholomorphic invariant introduced by B. Fridman on strongly pseudoconvex domains,…

We simplify proof of the theorem that close to any pseudoholomorphic disk there passes a pseudoholomorphic disk of arbitrary close size with any pre-described sufficiently close direction. We apply these results to the Kobayashi and Hanh…

The Ricci curvature of the Bergman metric on a bounded domain $D\subset \mathbb{C}^n$ is strictly bounded above by $n+1$ and consequently $\log (K_D^{n+1}g_{B,D})$, where $K_D$ is the Bergman kernel for $D$ on the diagonal and $g_{B, D}$ is…

Let n be a natural number equal or greater than 2. In this paper we study the topological structure of certain hyperspaces of convex subsets of constant width, equipped with the Hausdorff metric topology. We focus our attention on the…