follows that $(M,J,\theta)$ is the Sasakian space form with the positive Tanaka-Webster constant scalar curvature and vanishing torsion.

(proof of Theorem \[t4\] and Corollary \[c6\])

It follows from (\[2018BB\]) that $$\int_{M}\left(  R-\frac{1}{2}Tor-\frac{1}{2}Tor^{\prime}\right)  \left(
\gamma,\gamma \right)  d\mu+\int_{M}\left \vert \gamma_{1,1}\right \vert ^{2}d\mu+\int_{M}\left(  P_{0}u^{\bot}\right)  u^{\bot}d\mu=0$$ if $$Q=0.$$ Then we are again in the line of (\[2018BBB\]) and we have$$\gamma=0$$ under the assumptions. Again we have $$u^{\bot}=0$$ and by (\[0\]) $$R_{,1}-iA_{11},_{\bar{1}}=2P_{1}u_{\ker}.$$ Hence $$\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta=e^{\frac{\left(
f+2u_{\ker}\right)  }{3}}\theta$$ which is pseudo-Einstein. It follows from (\[0c\]) that $$P_{1}u_{\ker}=0$$ and then $$R_{,1}-iA_{11},_{\bar{1}}=0$$ which is pseudo-Einstein as well. The argumet for CR uniformization theorem as in Corollary \[c6\] are easily derived from the previous one. Then we are done.

Appendix
========

In this appendix, we introduce some basic notions from pseudohermitian geometry as in [@l].

\[def:CR\] Let $M$ be a smooth manifold and $\xi \subset TM$ a subbundle. A **CR structure** on $\xi$ consists of an endomorphism $J:\xi
\rightarrow \xi$ with $J^{2}=-id$ such that the following integrability condition holds.

1\. If $X,Y\in \xi$, then so is $[JX,Y]+[X,JY]$.

2\. $J([JX,Y]+[X,JY])=[JX,JY]-[X,Y]$.

The CR structure $J$ can be extended to $\xi \otimes{\mathbb{C}}$, which we can then decompose into the direct sum of eigenspaces of $J$. The eigenvalues of $J$ are $i$ 