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Let S ⊂ ℂⁿ, n ≥ 3, be a compact connected 2-codimensional submanifold having the following property: there exists a Levi-flat hypersurface whose boundary is S, possibly as a current. Our goal is to get examples of such S containing at least one special 1-hyperbolic point: a sphere with two horns, elementary models and their gluings. Some particular cases of S being a graph are also described.
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