Interpolation by proper holomorphic embeddings of the disc into \(\mathbb{C}^2\)
type: Item
instance of: scholarly article
title: Interpolation by proper holomorphic embeddings of the disc into \(\mathbb{C}^2\)
zbMATH Open document ID: 1026.32030
author: Josip Globevnik
published in: Mathematical Research Letters
publication date: 2003-11-25T00:00:00Z
review text: Let \(\Delta\) be the open unit disc in \(\mathbb C\). A map \(f: \Delta\mathbb\to \mathbb C^2\) is called a proper holomorphic embedding if it is a holomorphic immersion which is one to one and such that the preimage of every compact set is compact. It is known that given a discrete set \(E\subset\mathbb C^2\) there is a proper holomorphic embedding \(f:\Delta\to\mathbb C^2\) such that \(E\subset f(\Delta)\). The author proves a stronger result: Given a discrete set \(S\subset\Delta\) and a proper injection \(\varphi: S\to\mathbb C^2\) there is a proper holomorphic embedding \(f:\Delta\to\mathbb C^2\) that extends \(\varphi\). In other words, given an injective sequence \(\{\zeta_j\}\subset\Delta\) such that \(|\zeta_j|\to 1\) and an injective sequence \(\{w_j\}\subset\mathbb C^2\) such that \(|w_j|\to +\infty\) there is a proper holomorphic embedding \(f:\Delta\to\mathbb C^2\) such that \(f(\zeta_j) = w_j\), \(j\in\mathbb N\). The proof of the Carleman approximation theorem of Buzzard and Forstnerič can be adapted to prove such a result for proper holomorphic embeddings \(f:\mathbb C\to\mathbb C^2\). In the proof one uses the fact that \(\mathbb C\) admits particularly simple embeddings into \(\mathbb C^2\) of the form \(\zeta\to (\zeta,a(\zeta))\) where \(a\) is an entire function. There are no such embeddings for \(\Delta\) so a different proof is necessary in the considered case. In the induction step of the proof, the author uses simultaneous composition by automorphisms on the left and on the right, a novelty introduced by Buzzard and Forstnerič.
Mathematics Subject Classification ID: 32H02
Mathematics Subject Classification ID: 32Q40
Mathematics Subject Classification ID: 32H35
zbMATH DE Number: 1886051
zbMATH Keywords: proper holomorphic embeddings
zbMATH Keywords: proper injection
zbMATH Keywords: injective sequence
zbMATH Keywords: Carleman approximation
zbMATH Keywords: induction process
zbMATH Keywords: holomorphic automorphism
zbMATH Keywords: polynomially convex set
zbMATH Keywords: simultaneous composition by automorphisms
MaRDI profile type: Publication
full work available at URL: mrl.2002.v9.n4.a16
OpenAlex ID: W1984653425
DOI: 10.4310/MRL.2002.V9.N4.A16
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description: scientific article; zbMATH DE number 1886051