Locally determined envelopes of holomorphy in \({\mathbb{C}}\) 2
Let \(\Omega\) be a domain in \({\mathbb{C}}^ n.\) The envelope of holomorphy E(\(\Omega)\) of \(\Omega\) is said to be locally determined at \(p\in \partial \Omega\) if there exists a domain \({\tilde \Omega}\) (called a local envelope) such that \(p\in \partial {\tilde \Omega}\) and for every small ball U' with \(p\in U'\) there exists an open neighbourhood U'' of p for which \(U''\cap E(\Omega \cap U')={\tilde \Omega}\cap U''\). The author proves that: if \(\Omega\) \(\subset {\mathbb{C}}^ 2 \)is a smoothly bounded domain, \(0\in \partial \Omega\) and \(\partial \Omega\) near 0 has the form \[ v=\alpha uz\bar z+z^ 2\bar z^ 2+\mu re z\bar z^ 3+ \text{(terms of weight }\geq 5), \] with \({\mathbb{R}}\ni \alpha \neq 0\), \(0\leq \mu <4/3\) then the envelope of \(\Omega\) is locally determined at 0; the author presents also an effective construction and certain characterizations of \({\tilde \Omega}\).
scientific article; zbMATH DE number 4065402
scientific article