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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 $$\partial M=\Gamma$$.
##### MSC:
 32D10 Envelopes of holomorphy
##### Keywords:
envelope of holomorphy; analytic disc
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##### References:
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