On the envelope of holomorphy of a 2-sphere in \(\mathbb{C}^ 2\). (English) Zbl 0736.32009

Bedford, Eric; Klingenberg, Wilhelm

J. Am. Math. Soc. 4, No. 3, 623-646 (1991).

Let \(\Gamma\) be a smooth 2-sphere imbedded in a 2-dimensional Stein manifold. In order to investigate the envelope of holomorphy and the polynomial hull of a 2-sphere in \({\mathbb{C}}^ 2\), the authors study the problem of finding a 3-dimensional ball \(B\) such that \(\partial B=\Gamma\) and \(B\) is foliated by complex disks.
The main results are the following. Let \(\Omega\Subset X\) be a strongly pseudoconvex domain with smooth boundary and \(\Gamma\) smoothly imbedded in \(\partial\Omega\). There is a small \(C^ 2\) perturbation \(\Gamma'\) of \(\Gamma\) such that \(\Gamma'\) is swept out by the boundaries of complex disks. Further there is a smooth 3-ball \(B'\) that is the disjoint union of complex disks, such that \(\partial B'=\Gamma'\) and \(\bar B'\) is the envelope of holomorphy of \(\Gamma'\). Moreover, if the tangent planes at the points of \(\Gamma\) satisfy suitable technical conditions there is a 3-dimensional topological manifold \(B\), that is the disjoint union of complex disks, such that \(\partial B=\Gamma\) and \(\bar B\) is the envelope of holomorphy of \(\Gamma\).

Reviewer: A.Tancredi (Perugia)

MSC:

32D10	Envelopes of holomorphy
32E20	Polynomial convexity, rational convexity, meromorphic convexity in several complex variables

Keywords:

elliptic point; hyperbolic point; envelope of holomorphy; polynomial hull; 2-sphere