Discs in Stein manifolds containing given discrete sets
type: Item
instance of: scholarly article
title: Discs in Stein manifolds containing given discrete sets
zbMATH Open document ID: 1014.32015
author: Barbara Drinovec Drnovšek
published in: Mathematische Zeitschrift
publication date: 2002-06-27T00:00:00Z
review text: Let \(\Delta\) denote the unit disc in \({\mathbb C}.\) The following two results are known: i) Given a discrete subset \(S\) of a convex domain \(D \subset {\mathbb C}^N\) there exists a proper holomorphic map \(f\colon \Delta \rightarrow D\) such that \(S \subset f(D).\) ii) Given a Stein manifold \(M\) of dimension at least two, a point \(z \in M\) and a direction \(X \in T_zM - \{0\}\) there exists a proper holomorphic map \(f\colon \Delta \rightarrow M\) such that \(f(0)=z\) and \(f'(0)=\lambda X,\) with \(\lambda \neq 0.\) The author generalizes these two results by showing that, given a discrete subset \(\{z_n\}\) of a Stein manifold \(M\) with dim\(M \geq 2,\) there exists a proper holomorphic immersion \(f\colon \Delta \rightarrow M\) such that \(z_n \in f(\Delta)\) for all \(n,\) and with prescribed higher order contact with given one dimensional submanifolds of \(M\) at the points \(z_n.\) If \(\dim M \geq 3\) she shows that \(f\) can be chosen to be a proper embedding.
reviewed by: Elizabeth Gasparim
Mathematics Subject Classification ID: 32Q28
Mathematics Subject Classification ID: 32Q40
zbMATH DE Number: 1761070
zbMATH Keywords: Stein manifolds
zbMATH Keywords: proper embedding
MaRDI profile type: Publication
full work available at URL: s002090100320
OpenAlex ID: W1992594486
DOI: 10.1007/S002090100320
description: scientific article; zbMATH DE number 1761070