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Global convexity properties of some families of three-dimensional compact Levi-flat hypersurfaces. (English) Zbl 0761.32010
Assume, that on a complex manifold $$M$$ a real analytic hypersurface $$S$$ is given together with a foliation $$F$$ of $$S$$ into complex hypersurfaces (relative $$M$$). One may expect some interplay between properties of the foliation and the way, $$S$$ is embedded in $$M$$ (properties of neighbourhoods of $$S$$ in $$M$$). The author tries to attack this problem at the hand of two quite special types in the case $$\dim_ \mathbb{C} M=2$$, $$S$$ compact (extending also an example of Ohsawa). One type is of the form $$S=T^ 3$$ (torus) with some special real analytic CR-structure, and $$M$$ being the complexification of this. He proves, that five (respectively six) of certain properties of $$F$$ and $$S\subset M$$ are equivalent. He also points out that in different situations, even if $$dim_ \mathbb{C} M=2$$, these equivalences do not hold in general. And in cases $$dim_ \mathbb{C} M>2$$ the situation still may be much more involved. The referee proposes, also to take the leaf-structure $$S/F$$ into account.
Reviewer: K.Spallek (Bochum)

MSC:
 32V40 Real submanifolds in complex manifolds 32T99 Pseudoconvex domains 57R30 Foliations in differential topology; geometric theory
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References:
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