A fourth order curvature flow on a CR 3-manifold Shu-Cheng ChangJih-Hsin ChengHung-Lin Chiu 32V2053C44CR manifoldTanaka-Webster curvaturetorsionMoser inequality$Q$-curvature flowPaneitz operatorSub-LaplacianKohn Laplacian Let $(\mathbf{M}^3, J, \theta_0)$ be a closed pseudohermitian $3$-manifold. Suppose the associated torsion vanishes and the associated $Q$-curvature has no kernel part with respect to the associated Paneitz operator. On such a background pseudohermitian $3$-manifold, we study the change of the contact form according to a certain version of normalized $Q$-curvature flow. This is a fourth order evolution equation. We prove that the solution exists for all time and converges smoothly to a contact form of zero $Q$-curvature. We also consider other background conditions and obtain a priori bounds on high-order norms on the solutions. Indiana University Mathematics Journal 2007 text pdf 10.1512/iumj.2007.56.3001 10.1512/iumj.2007.56.3001 en Indiana Univ. Math. J. 56 (2007) 1793 - 1826 state-of-the-art mathematics http://iumj.org/access/