\[r2\], $M$ admits a Riemannian metric of positive scalar curvature if the pseudohermitian curvature is $\left(  \frac{1}{2},\frac{1}{2}\right)  $-positive. Then the properties of pseudohermitian curvature positivity and nonnegative CR Paneitz operator $P_{0}$ imply the embeddability of $(M,J,\theta_{0})$ in the complex Euclidean space $\mathbf{C}^{N}$ ([@ccy]). Furthermore, the Paneitz operator $P_{0}$ with respect to $(J,\theta)$ is essentially positive in this special case as well.

Combining Theorem \[t3\], Corollary \[c3\], (\[0\]) and the above remark, we have the following CR uniformization theorem ([@t]) in a Sasakian manifold due to the eigenvalue estimate of the CR Paneitz operator (\[2018H\]).

\[c4\] If $(M,J,\theta)$ is a closed strictly pseudoconvex CR $3$-manifold with $c_{1}(T_{1,0}M)=0$. Assume that the manifold is Sasakian and the Tanaka-Webster scalar curvature is positive, then $$\Lambda^{2}\int_{M}(u^{\bot})^{2}d\mu \leq \int_{M}(Q^{\bot})^{2}d\mu.$$ In additional, if the CR  $Q$-curvature is pluriharmonic (i.e. $Q^{\bot}=0),$ then $(M,J,\theta)$ is the Sasakian space form with the positive constant Tanaka-Webster scalar curvature and vanishing torsion.

Finally, if we do not assume the torsion is vanishing (non-Sasakian), we can derive another existence theorem for the pseudo-Einstein contact form with the stronger condition.

\[t3a\] 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 