nonnegative pseudohermitian Ricci curvature tensor or the vanishing pseudohermitian torsion. However, his method couldn’t be applied to the case $n=1$ directly.

Then it is natural to focus on such existence theorem of pseudo-Einstein contact forms for $n=1$. Of course, we must find another appropriate definition for the pseudo-Einstein contact form. In fact by Lemma \[l1\] below, it is reasonable to view$$W_{1}\doteqdot \left(  R,_{1}-iA_{11},_{\overline{1}}\right)  =0$$ as the pseudo-Einstein contact form in a closed strictly pseudoconvex CR $3$-manifold $\left(  M,J,\theta \right)  $.

Before we start to work on the existence of pseudo-Einstein contact forms in a closed strictly pseudoconvex CR $3$-manifold, we make the following observation in a closed strictly pseudoconvex CR $(2n+1)$-manifold $(M,J)$ with a choice of pseudohermitian contact form $\theta$.

\(i) For $n\geq2$ : Assume that the pseudohermitian Ricci curvature is positive, it is well-known ([@k], [@l]) that we have the solvability of the inhomogeneous tangential Cauchy-Riemann equation

$$\overline{\partial}_{b}\varphi=\eta \label{2018B}$$

for any $\overline{\partial}_{b}$-closed $(0,1)$-form $\eta.$ That is to say that

$$H_{\overline{\partial}_{b}}^{0,1}(M)=0.$$

\(ii) For $n=1:$ We consider an embeddable closed strictly pseudoconvex CR $3$-manifold $(M,\theta)$ with $c_{1}(T_{1,0}M)=0$. There is a pure imaginary $1$-form$$\sigma=\sigma_{\overline{1}}\theta^{\overline{1}}-\sigma_{1}\theta^{1}+i\sigma_{0}\theta \label{25a}$$ such that $$d\omega_{1}^{1}=d\sigma.$$ Due to J. J. Kohn’s result (Lemma \[l3\]), we observe that there are a 