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Envelope of holomorphy of certain two-manifolds in \(\mathbb C^{2}\). (English) Zbl 1220.32001
Let \(\Gamma\) be a two-sphere imbedded to a boundary \(\partial\Omega\) of a bounded pseudoconvex domain \(\Omega\subset\mathbb{C}^2\). Suppose that \(\partial\Omega\) is real analytic and \(\Gamma\) has only isolated points with complex tangencies. In a neighborhood of such point \(\Gamma\) can be written as \[ \omega= |z|^{2k}+\lambda\text{\,Re}(z^{2k})+ E_{2k}(z,\overline z)+ O(|z|^{2k+1}), \] where \(E_{2k}(z,\overline z)= \sum_{\substack{ m+n= 2k\\ n\neq m}} a_{m,n} z^m\overline z^n\). A point is said of elliptic type if \[ \lambda+ \sum_{m<n} \Biggl(2-{m\over k}\Biggr)|a_{m,n}|+ \sum_{m> n} |a_{m,n}|< 1 \] and of hyperbolic type if \(\lambda> 1\). The main result of the paper is the following
Theorem. In the conditions as above suppose that all points with complex tangencies are
\(\bullet\) either of elliptic type,
\(\bullet\) or if hyperbolic type such that \(\sum|a_{m,n}|< \min\{{|\lambda-1|\over 3},{1\over 6}\}\).
Then the envelope of holomorphy of \(\Gamma\) is a smooth three-manifold \(M\) foliated by analytic discs and \(\partial M=\Gamma\).
32D10 Envelopes of holomorphy
Full Text: DOI
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