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On boundaries of Levi-flat hypersurfaces in \(\mathbb C^n\). (English. Abridged French version) Zbl 1085.32019
Given a \(2\)-codimensional real compact submanifold \(S\) of \(\mathbb{C}^n\), \(n>2\), the authors try to construct a compact hypersurface \(M\), with boundary \(S\), and \(M\setminus S\) Levi-flat. They make the following assumptions on \(S\): (i) \(S\) is minimal at every CR point; (ii) every complex point of \(S\) is flat and elliptic and \(S\) contains at least one point with this property; (iii) \(S\) does not contain complex submanifolds of complex dimension \((n-2)\). Then they show that there exists a Levi-flat real hypersurface \(\tilde{M}\subset\mathbb{C}\times\mathbb{C}^n\) with negligible singularities and boundary \(\widetilde{S}\) such that the natural projection \(\pi:\mathbb{C}\times\mathbb{C}^n\rightarrow \mathbb{C}^n\) restricts to a CR diffeomorphism of \(\widetilde{S}\) onto \(S\).

32V25 Extension of functions and other analytic objects from CR manifolds
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