Some Q-curvature Operators on Five-Dimensional Pseudohermitian Manifolds

We construct Q-curvature operators on d-closed (1, 1)-forms and on ∂¯b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\partial }_b$$\end{document}-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 ∂¯b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\partial }_b$$\end{document}-closed (0, 1)-forms is nonnegative.