in this joint work with pierre dolbeault and giuseppe tomassini we consider the problem of characterizing compact real submanifolds of $c^n$ that bound levi flat hypersurfaces. the problem is well studied in $c^2$ but surprisingly little is known in higher dimension. in this talk i will, in particular, explain the fundamental difference between $n=2$ and higher dimension showing why the known methods in $c^2$ do not apply.