functions. Assume that  the pseudohermitian curvature is $\left(  \frac{1}{2},\frac{1}{2}\right)
$-positive $$R>(|A_{11}|+\left \vert A_{11,\overline{1}}\right \vert ).$$ If the CR  $Q$-curvature is pluriharmonic, then $\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$ is a pseudo-Einstein contact form for any CR-pluriharmonic function $f$ with $$\gamma=0.$$

As a consequence, we have the CR uniformization theorem ([@t]) in a spherical CR $3$-manifold.

\[c5\] Let $(M,J,\theta)$ be a closed strictly pseudoconvex CR $3$-manifold of $c_{1}(T_{1,0}M)=0$ and nonnegative CR Paneitz operator $P_{0}$ with kernel consisting of the CR-pluriharmonic functions. Assume that  the pseudohermitian curvature is $\left(  \frac{1}{2},\frac{1}{2}\right)
$-positive and the CR  $Q$-curvature is pluriharmonic, then $(M,J,\theta)$ is a pseudo-Einstein CR manifold. In additional, if it is spherical with positive constant Tanaka-Webster scalar curvature, then $(M,J,\theta)$ is the Sasakian space form with the positive constant Tanaka-Webster scalar curvature and vanishing torsion.

In the course of the proof of Theorem \[t3a\], we assume that the CR  $Q$-curvature is pluriharmonic. However, in particular for assuming vanishing of the CR $Q$-curvature *(see section* $2$*)*, we can trop the condition of (\[2018C\]) as the following :

\[t4\] Let $(M,J,\theta)$ is a closed strictly pseudoconvex CR $3$-manifold of $c_{1}(T_{1,0}M)=0$ with nonnegative CR Paneitz operator $P_{0}. $ Assume that  the pseudohermitian curvature is $\left(  \frac{1}{2},\frac{1}{2}\right)  $-positive $$R>(|A_{11}|+\left \vert A_{11,\overline{1}}\right 