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.