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

TL;DR

This paper introduces new $Q$-curvature operators on five-dimensional pseudohermitian manifolds, providing a novel formula for scalar $Q$-curvature and applications to characterizing CR manifolds with flat contact forms.

Contribution

It constructs specific $Q$-curvature operators on forms and derives a new scalar $Q$-curvature formula, with applications to CR geometry and flat contact forms.

Findings

Cohomological characterization of CR five-manifolds with $Q$-flat contact forms

Existence of $Q$-flat contact forms under nonnegativity conditions

New formula relating $Q$-curvature to geometric structures

Abstract

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.