Approximation of biholomorphic mappings by automorphisms of \(\mathbb{C}^ n\)
type: Item
instance of: scholarly article
title: Approximation of biholomorphic mappings by automorphisms of \(\mathbb{C}^ n\)
zbMATH Open document ID: 0792.32011
DOI: 10.1007/BF01232438
author: Franc Forstnerič
author: Jean-Pierre Rosay
published in: Inventiones Mathematicae
publication date: 1994-07-28T00:00:00Z
full work available at URL: https://eudml.org/doc/144107
review text: Let \(\Omega\) be open in \(\mathbb{C}^ n\) and \(\varphi_ t:\Omega \to\varphi_ t (\Omega) \subset \mathbb{C}^ n\), \(0 \leq t \leq 1\), a smooth family of biholomorhic mappings. The authors discuss at first the problem of sufficient (and necessary) conditions for an approximation of these mappings (uniformly on compact subsets of \(\Omega)\) by holomorphic automorphisms of \(\mathbb{C}^ n\). They recall (a generalized version of) a theorem of \textit{E. Andersén} and \textit{L. Lempert} [Invent. Math. 110, 371-388 (1992; Zbl 0770.32015)]: If every \(\varphi_ t (\Omega)\) is Runge in \(\mathbb{C}^ n\), then \(\varphi_ t\) can be approximated for every \(t \in[0,1]\) if and only if \(\varphi_ 0\) can be approximated. Conversely, if \(\Omega\) is a pseudoconvex Runge domain in \(\mathbb{C}^ n\) and the biholomorphic map \(\varphi:\Omega \to \varphi(\Omega) \subset \mathbb{C}^ n\) can be approximated by automorphisms of \(\mathbb{C}^ n\), then every compact subset \(K \subset \Omega\) has a neighborhood \(D\) and a smooth family \(\varphi_ t:D \to \varphi_ t (D) \subset \mathbb{C}^ n\) of biholomorphic mappings with \(\varphi_ 0=\text{id}_ D\), \(\varphi_ 1=\varphi\) and \(\varphi_ t(D)\) Runge in \(\mathbb{C}^ n\) for \(0 \leq t \leq 1\) (Theorem 1.1.). Using the fact that every polynomially convex subset \(K\) of \(\Omega\) has a basis of Stein neighborhoods that are Runge in \(\mathbb{C}^ n\), the authors show that for \(K\) compact and \(\varphi_ 0=\text{id}\) the image \(\varphi_ t (K)\) is polynomially convex, \(0 \leq t \leq 1\), if and only if \(K\) has a neighborhood \(U \subset \Omega\) with \(\varphi_ t (U)\) Runge in \(\mathbb{C}^ n\) for \(t \in[0,1]\) (Theorem 2.1). Now take two real-analytic submanifolds \(M_ 0\) and \(M_ 1\) of \(\mathbb{C}^ n\) and assume that they are diffeomorphic. When are \(M_ 0\) and \(M_ 1\) \(\mathbb{C}^ n\)-equivalent, i.e. when exists a biholomorphic map \(\varphi:\Omega \to \mathbb{C}^ n\) of a neighborhood \(\Omega\) of \(M_…
Mathematics Subject Classification ID: 32E30
Mathematics Subject Classification ID: 32E20
Mathematics Subject Classification ID: 32V40
zbMATH DE Number: 447106
zbMATH Keywords: biholomorhic mappings
zbMATH Keywords: approximation
zbMATH Keywords: holomorphic automorphisms
reviewed by: Eberhard Oeljeklaus
MaRDI profile type: Publication
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OpenAlex ID: W2122623998
description: scientific article; zbMATH DE number 447106