Some remarks concerning holomorphically convex hulls and envelopes of holomorphy
type: Item
instance of: scholarly article
title: Some remarks concerning holomorphically convex hulls and envelopes of holomorphy
zbMATH Open document ID: 0816.32011
DOI: 10.1007/BF02571894
author: Burglind Jöricke
published in: Mathematische Zeitschrift
publication date: 1995-07-23T00:00:00Z
full work available at URL: https://eudml.org/doc/174719
review text: Let \(\Omega\) be a bounded strictly convex domain in \(\mathbb{C}^ 2\) with smooth boundary, and let \(K\) be a compact subset of the boundary \(\partial \Omega\). Stout, Lupacciolu and Alexander proved that each continuous \(CR\)-function on \(\partial \Omega \backslash K\) has uniquely determined analytic extension to \(\Omega \backslash \widehat K\), \(\widehat K\) being the polynomial hull of \(K\). There is a heuristic way to see this, namely by observing the connection between Oka's characterization principle for polynomial hulls on the one hand and analytic continuation along continuous one-parameter families of one-dimensional analytic varieties on the other hand. Monodromy considerations for proving that the envelope of holomorphy is schlicht are based on the fact that in \(\mathbb{C}^ 2\) the complex codimension of the considered varieties is equal to one. These observations are managed into a rigorous proof, which differs from the original one. The result is generalized to arbitrary domains of holomorphy with \(C^ 2\) boundary, the polynomial hull in that case to be replaced by the hull with respect to the space of functions which are holomorphic in \(\Omega\) and continuous in the closure \(\overline \Omega\). Previous results of this kind concern the case of domains \(\Omega\) with \(\overline \Omega\) having a Stein neighbourhood basis and the (in general larger) hull with respect to functions which are analytic in a neighbourhood of \(\overline \Omega\).
Mathematics Subject Classification ID: 32D10
Mathematics Subject Classification ID: 32E20
Mathematics Subject Classification ID: 32D20
Mathematics Subject Classification ID: 32V05
Mathematics Subject Classification ID: 32E35
zbMATH DE Number: 737166
zbMATH Keywords: polynomial convexity
zbMATH Keywords: holomorphically convex hulls
zbMATH Keywords: envelope of holomorphy
zbMATH Keywords: Stein neighbourhood basis
MaRDI profile type: Publication
cites work: A Note on Polynomial Hulls
cites work: A note on Hulls
cites work: Boundary behavior of holomorphic functions on pseudoconvex domains
cites work: Q5615632
cites work: Q5542941
cites work: Spectre de \(A(\bar\Omega)\) pour des domaines bornes faiblement pseudoconvexes réguliers
cites work: Q5330830
cites work: Q5512342
cites work: A theorem on holomorphic extension of CR-functions
cites work: Rado's Theorem for CR-Functions
cites work: Polynomially and rationally convex sets
cites work: Uniform approximation on smooth curves
cites work: Analytic continuation and boundary continuity of functions of several complex variables
cites work: Q4694622
cites work: Orbits of Families of Vector Fields and Integrability of Distributions
cites work: Q1071902
OpenAlex ID: W2026369152
Recommended article: Q3490267
Recommended article: A note on envelopes of holomorphy
Recommended article: Envelopes of holomorphy and polynomial hulls
Recommended article: Some envelopes of holomorphy
Recommended article: Q3829778
Recommended article: POLYNOMIAL HULLS AND ENVELOPES OF HOLOMORPHY OF SUBSETS OF STRICTLY PSEUDOCONVEX BOUNDARIES
Recommended article: Q4281334
Recommended article: Q3973857
Recommended article: Q4937828
Recommended article: Q4204409
description: scientific article; zbMATH DE number 737166