  The complex Plateau problem is analogous, in a Hermitian complex manifold, to
the classical Plateau problem in 3 dimensional real space: it is a geometrical
problem of extension of a closed real manifold into a complex analytic
subvariety, or into a Levi-flat subvariety. Minimality of complex analytic
subvarieties and analogous properties of Levi-flat subvarieties, in K\"ahler
manifolds, are recalled or given. Known results in n dimensional complex and
projective complex spaces are recalled. Extensions to real parametric problems
are solved or proposed, leading to the construction of Levi-flat hypersurfaces
with prescribed boundary in some complex manifolds.
