Abstract
At the end of Chapter I and in Chapter III we met open sets in ℂn on which any holomorphic function can be extended to a larger open set. The open sets which do not have this property are called domains of holomorphy: in this chapter we study such open sets. We start by giving a characterisation of domains of holomorphy in terms of holomorphic convexity (the Cartan–Thullen theorem). We then introduce the notion of pseudoconvexity in order to get a more analytic characterisation of domains of holomorphy. This requires us to define plurisubharmonic functions. We then prove that every domain of holomorphy is pseudoconvex: the converse, which is known as the Levi problem, is studied in Chapter VII.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer-Verlag London Limited
About this chapter
Cite this chapter
Laurent-Thiébaut, C. (2011). VI Domains of holomorphy and pseudoconvexity. In: Holomorphic Function Theory in Several Variables. Springer, London. https://doi.org/10.1007/978-0-85729-030-4_6
Download citation
DOI: https://doi.org/10.1007/978-0-85729-030-4_6
Published:
Publisher Name: Springer, London
Print ISBN: 978-0-85729-029-8
Online ISBN: 978-0-85729-030-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)