A fourth order curvature flow on a CR 3-manifold

Abstract

Let (M 3 , J, θ 0 ) be a closed pseudohermitian 3-manifbld. 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.