The symmetrized polydisc cannot be exhausted by domains biholomorphic to convex domains

Exhausting domains of the symmetrized bidisc

We show that the symmetrized bidisc may be exhausted by strongly linearly convex domains. It shows in particular the existence of a strongly linearly convex domain that cannot be exhausted by domains biholomorphic to convex ones.

An example of a bounded $\Bbb C$-convex domain which is not biholomoprhic to a convex domain

We show that the symmetrized bidisc is a $\Bbb C$-convex domain. This provides an example of a bounded $\Bbb C$-convex domain which cannot be exhausted by domains biholomorphic to convex domains.

Lempert Theorem for stronly linearly convex domains with smooth boundaries

The aim of this paper is to present a detailed and slightly modified version of the proof of the Lempert Theorem in the case of non-planar stronlgy linearly convex domains with C^2 smooth boundaries. The original Lempert's proof is presented only in proceedings of a conference with a very limited access and at some places it was quite sketchy. We were encouraged by some colleagues to prepare an extended version of the proof in which all doubts could be removed and some of details of the proofs could be simplified. We hope to have done it below. Certainly the idea of the proof belongs entirely …

Limits of an increasing sequence of complex manifolds

Let $M$ be a complex manifold which admits an exhaustion by open subsets $M_j$ each of which is biholomorphic to a fixed domain $Ω\subset \mathbb C^n$. The main question addressed here is to describe $M$ in terms of $Ω$. Building on work of Fornaess--Sibony, we study two cases namely, $M$ is Kobayashi hyperbolic and the other being the corank one case in which the Kobayashi metric degenerates along one direction. When $M$ is Kobayashi hyperbolic, its complete description is obtained when $Ω$ is one of the following domains -- (i) a smoothly bounded Levi corank one domain, (ii) a smoothly bound…

Poincaré关于区域双全纯不等价定理的度量版本

A metric version of Poincaré's theorem concerning biholomorphic inequivalence of domains

本文推广了Poincaré定理，证明在Kobayashi距离下，当因子数更多时，凸有界域乘积无法等距嵌入强凸域乘积，并基于度量几何建立了一般性等距嵌入障碍准则。

On isometries of the Kobayashi and Carathéodory metrics

This article considers isometries of the Kobayashi and Carathéod-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 Poincaré's theorem about biholomorphic inequivalence of $ \mathbf{B}^n $, the unit ball in $ \mathbf{C}^n $ and $ Δ^n $, the unit polydisc in $ \mathbf{C}^n $ and then provide few examples which \textit{suggest} that $ \mathbf{B}^n $ cannot be mapped isometrically onto a product domain. In addition, we prove several results on continuous extension of isometries $ f : D_1 \rightarrow D_2 $ to…