Schlicht envelopes of holomorphy and foliations by lines
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.
Keywords
Cite
@article{arxiv.0802.0727,
title = {Schlicht envelopes of holomorphy and foliations by lines},
author = {Finnur Larusson and Rasul Shafikov},
journal= {arXiv preprint arXiv:0802.0727},
year = {2008}
}
Comments
Version 2: Terminology revised and title changed in view of information about the origins of what we now call "the schlichtness lemma" that we didn't have when we finished version 1. Version 3: A few minor changes. To appear in Journal of Geometric Analysis