Some $Q$-curvature operators on five-dimensional pseudohermitian manifolds

We construct $Q$-curvature operators on $d$-closed $(1,1)$-forms and on $overline{partial}_b$-closed $(0,1)$-forms on five-dimensional pseudohermitian manifolds. These closely related operators give rise to a new formula for the scalar $Q$-curvature. As applications, we give a cohomological characterization of CR five-manifolds which admit a $Q$-flat contact form; and we show that every closed, strictly pseudoconvex CR five-manifold with trivial first real Chern class admits a $Q$-flat contact form provided the $Q$-curvature operator on $overline{partial}_b$-closed $(0,1)$-forms is nonnegative.