On boundaries of Levi-flat hypersurfaces in C n

Let S be a smooth 2-codimensional real compact submanifold of C n , n > 2 . We address the problem of finding a compact hypersurface M , with boundary S , such that M ∖ S is Levi-flat. We prove the following theorem. Assume that (i) S is nonminimal at every CR point, (ii) every complex point of S is flat and elliptic and there exists at least one such point, (iii) S does not contain complex submanifolds of dimension n − 2 . Then there exists a Levi-flat ( 2 n − 1 ) -subvariety M ˜ ⊂ C × C n with negligible singularities and boundary S ˜ (in the sense of currents) such that the natural projection π : C × C n → C n restricts to a CR diffeomorphism between S and S ˜ .