On Levi-flat hypersurfaces with given boundary in \(\mathbb C^n\)
The author discusses the following problem. Consider a compact connected \(2\)-codimensional submanifold \(S\subset {\mathbb{C}}^n \) and find a Levi flat hypersurface \(M\subset {\mathbb{C}}^n\setminus S\) such that \(dM=S\) i.e. \(M\) has the boundary \(S\)(possibly as current). The author studies the behaviour at \(CR\)-points, at complex points and elliptic or hyperbolic flat points. Then he studies the induced foliation by \(CR\)-orbits in the case of elliptic flat points and in the case of hyperbolic flat points. He considers the boundary value problem for embedded surfaces, studying the subvarieties with negligible singularities and treating the same boundary problem in a real hyperplane of \({\mathbb{C}}^n\). Finally, he presents the problem of Levi flat \((2n-1)\)-subvarieties with a given boundary in \({\mathbb{C}}^n\). The case where \(n=2\) is considered separately.
scientific article; zbMATH DE number 5323338