2008/02/06 by Finnur Lárusson, Finnur Larusson, Larusson, Finnur +2
Mathematics · #32A10 #32A40 #32D10 (Primary) #32D15 #32M25 #32Q28 #32S25 #37F75 (Secondary) #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:32A10 #msc:32A40 #msc:32D10 #msc:32D15 #msc:32M25 #msc:32Q28 #msc:32S25 #msc:37F75
paper · pdf · doi:10.48550/arxiv.0802.0727
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
openalex publication_date 2008/02/06 · arxiv created 2008/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.