Global existence and convergence for the CR Q-curvature flow in a closed strictly pseudoconvex CR 3-manifold

In this note, we affirm the partial answer to the long open Conjecture which states that any closed embeddable strictly pseudoconvex CR $3$-manifold admits a contact form $\theta $ with the vanishing CR $Q$-curvature. More precisely, we deform the contact form according to an CR analogue of $Q$%-curvature flow in a closed strictly pseudoconvex CR $3$-manifold $(M,\ J,[\theta_{0}])$ of the vanishing first Chern class $c_{1}(T_{1,0}M)$. Suppose that $M$ is embeddable and the CR Paneitz operator $P_{0}$ is nonnegative with kernel consisting of the CR pluriharmonic functions. We show that the solution of CR $Q$-curvature flow exists for all time and has smoothly asymptotic convergence on $M\times \lbrack 0,\infty ).$\ As a consequence, we are able to affirm the Conjecture in a closed strictly pseudoconvex CR $3$-manifold of the vanishing first Chern class and vanishing torsion.