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https://hdl.handle.net/2440/51559
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Type: | Journal article |
Title: | Schlicht Envelopes of Holomorphy and Foliations by Lines |
Author: | Larusson, F. Shafikov, R. |
Citation: | Journal of Geometric Analysis, 2009; 19(2):373-389 |
Publisher: | Mathematica Josephina Inc |
Issue Date: | 2009 |
ISSN: | 1050-6926 1559-002X |
Statement of Responsibility: | Finnur Lárusson and Rasul Shafikov |
Abstract: | Given a domain Y in a complex manifold X, it is a difficult problem with no general solution to determine whether Y has a schlicht envelope of holomorphy in X, and if it does, to describe the envelope. The purpose of this paper is to tackle the problem with the help of a smooth 1-dimensional foliation F of X with no compact leaves. We call a domain Y in X an interval domain with respect to F if Y intersects every leaf of F in a nonempty connected set. We show that if X is Stein and if F satisfies a new property called quasiholomorphicity, then every interval domain in X has a schlicht envelope of holomorphy, which is also an interval domain. This result is a generalization and a global version of a well-known lemma from the mid-1980s. We illustrate the notion of quasiholomorphicity with sufficient conditions, examples, and counterexamples, and present some applications, in particular to a little-studied boundary regularity property of domains called local schlichtness. © 2009 Mathematica Josephina, Inc. |
Keywords: | Envelope of holomorphy Schlicht envelope of holomorphy Analytic continuation Stein manifold Foliation Quasiholomorphic Interval domain Locally schlicht |
Description: | The original publication can be found at www.springerlink.com |
DOI: | 10.1007/s12220-008-9058-3 |
Published version: | http://dx.doi.org/10.1007/s12220-008-9058-3 |
Appears in Collections: | Aurora harvest 5 Mathematical Sciences publications |
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