---
abstract: 'In this note, we mainly focus on the existence of pseudo-Einstein contact forms, an upper bound eigenvalue estimate for the CR Paneitz operator and its applications to the uniformization theorem for Sasakian space form in an embeddable closed strictly pseudoconvex CR $3$-manifold. Firstly, the existence of pseudo-Einstein contact form is confirmed if the CR $3$-manifold is Sasakian. Secondly, we derive an eigenvalue upper bound estimate for the CR Paneitz operator and obtain the CR uniformization theorem for a class of CR $3$-manifolds. At the end, under the positivity assumption of the pseudohermitian curvature, we derive the existence theorem for pseudo-Einstein contact forms and uniformization theorems in a closed strictly pseudoconvex CR $3$-manifold of nonnegative CR Paneitz operator with kernel consisting of the CR-pluriharmonic functions and the CR $Q$-curvature is CR-pluriharmonic.'
address:
- 'Department of Mathematics and Taida Institute for Mathematical Sciences (TIMS), National Taiwan University, Taipei 10617, Taiwan, R.O.C.'
- 'Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan '
- 'Yau Mathematical Sciences Center, Tsinghua University, Haidian District, Beijing 100084, China'
author:
- 'Shu-Cheng Chang'
- 'Ting-Jung Kuo'
- Chien Lin
title: 'Pseudo-Einstein structure, eigenvalue estimate for the CR Paneitz operator and its applications to uniformization theorem '
---

Introduction
============

 In Riemannian geometry, a Riemannian manifold is called Einstein if the Ricci curvature tensor is function-proportional to its Riemannian metric. For dimension greater than $2$, it is equivalent to the constant-proportional case. In the contrast to the Riemannian geometry situation, there is a resembling notion that a strictly pseudoconvex CR $(2n+1)$-manifold is called pseudo-Einstein if the pseudohermitian Ricci curvature tensor is function-proportional to its Levi metric. The pseudo-Einstein condition is less rigid than the Einstein condition in Riemannian geometry. Indeed, the CR contracted Bianchi identity no longer implies the pseudohermitian scalar curvature $R$ to be a constant due to the presence of pseudohermitian torsion for $n\geq2$ $$R_{\alpha \overline{\beta}}{}_{,\beta}=R_{\alpha}-i(n-1)A_{\alpha \beta}{}_{,\overline{\beta}}.$$ Note that any contact form on a closed strictly pseudoconvex $3$-manifold is actually pseudo-Einstein since the pseudohermitian Ricci tensor has only one component $R_{1\overline{1}}$.

In [@l], J. Lee showed that the obstruction to the existence of a pseudo-Einstein contact form $\theta$ is that its first Chern class $c_{1}(T_{1,0}M)$ vanishes. Indeed, for a closed strictly pseudoconvex $(2n+1)
$-manifold $\left(  M,J,\theta \right)  $ with $c_{1}(T_{1,0}M)=0$ and $n\geq2
$, he proved that $M$ admits a globally defined pseudo-Einstein contact form if either $M$ admits a contact form $\theta$ with 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 complex function $$\varphi=u+iv\in C_{\mathbb{C}
}^{\infty}\left(  M\right)$$ and $\gamma=\gamma_{\overline{1}}\theta^{\overline{1}}\in \Omega^{0,1}\left(
M\right)  \cap \ker \left(  \square_{b}\right)  $ such that$$\overline{\partial}_{b}\varphi=\sigma_{\overline{1}}\theta^{\overline{1}}-\gamma_{\overline{1}}\theta^{\overline{1}}\label{2018}$$ with $$\overline{\partial}_{b}(\sigma_{\overline{1}}\theta^{\overline{1}})=0.$$ Here $\square_{b}=2\left(  \overline{\partial}_{b}\overline{\partial}_{b}^{\ast}+\overline{\partial}_{b}^{\ast}\overline{\partial}_{b}\right)  $ is the Kohn-Rossi Laplacian. Then it is the natural question when we have the solvability of the inhomogeneous tangential Cauchy-Riemann equation (i.e. $\gamma=0$) $$\overline{\partial}_{b}\varphi=\sigma_{\overline{1}}\theta^{\overline{1}}.\label{2018A}$$

In this paper, we mainly focus on the existence of pseudo-Einstein contact forms as in Corollary \[c2\], Theorem \[t2\], Theorem \[t3a\] and Theorem \[t4\], an upper bound eigenvalue estimate for the CR Paneitz operator as in Theorem \[t3\], Corollary \[c4\] and its applications to the uniformization theorem for Sasakian space form as in Corollary \[c4\], Corollary \[c5\] and Corollary \[c6\] in a closed strictly pseudoconvex CR $3$-manifold.

We first state one of the main theorems as follows:

\[t1\]If $(M,J,\theta)$ is an embeddable closed strictly pseudoconvex CR $3$-manifold with $c_{1}(T_{1,0}M)=0$.  Then

\(i) $\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$ is a pseudo-Einstein contact form if and only if $f$ satisfies the third-order partial differential equation$$P_{1}f=i\left(  A_{11}\gamma_{\overline{1}}-\gamma_{1,0}\right)  .\label{7}$$ Here $P_{1}$ is a third-order CR-pluriharmonic operator $$P_{1}f=f_{\overline{1}11}+iA_{11}f_{\overline{1}}.$$ (ii) In particular, $\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$ is a pseudo-Einstein contact form for a CR-pluriharmonic function $f$ if and only if the equality holds $$\left(  A_{11}\gamma_{\overline{1}}-\gamma_{1,0}\right)  =0.\label{13}$$

In [@le], Lempert showed that a closed strictly pseudoconvex CR $3$-manifold $(M,J,\theta)$ with vanishing pseudohermitian torsion is embeddable. As a consequence, we are able to show that one of existence theorems for the pseudo-Einstein contact form in this paper.

\[c2\] Let $(M,J,\theta)$ be a closed strictly pseudoconvex CR $3$-manifold with $c_{1}(T_{1,0}M)=0$. Then $M$ admits a globally defined pseudo-Einstein contact form $e^{\frac{\left(  f+2u\right)  }{3}}\theta$ for any CR-pluriharmonic function $f$ if the pseudohermitian torsion is vanishing (i.e. Sasakian). More precisely, we have $$\gamma_{1,0}=0.$$

Note that we do not know whether it holds that $\gamma=0$ in the situation as in Corollary \[c2\]. However, by deriving the Bôchner-type estimate as in the Lemma \[l8\], we have$${\displaystyle \int \limits_{M}}
\left(  2R-Tor\right)  \left(  \gamma,\gamma \right)  d\mu+2{\displaystyle \int \limits_{M}}
\left \vert \gamma_{1,1}\right \vert ^{2}d\mu+\frac{1}{2}{\displaystyle \int \limits_{M}}
\left(  P_{0}f\right)  fd\mu=0$$ if $\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$ is a pseudo-Einstein contact form. It will conclude $$\gamma=0$$ under certain pseudohermitian geometric assumptions and obtain the solvability of the inhomogeneous tangential Cauchy-Riemann equation (\[2018A\]). Hereafter, we will use the result in [@ccy] implicitly, which says that if $(M,J,\theta)$ is a closed strictly pseudoconvex CR $3$-manifold with nonnegative CR Paneitz operator and positive Tanaka-Webster scalar curvature, then $(M,J,\theta)$ is embeddable. This ensures the embeddability in the hypothesis holds for closed strictly pseudoconvex CR $3$-manifolds.

\[t2\] If $(M,J,\theta)$ is a closed strictly pseudoconvex CR $3$-manifold with $c_{1}(T_{1,0}M)=0$ and nonnegative CR Paneitz operator $P_{0} $. Assume that the pseudohermitian curvature is $\left(  \frac{1}{2},0\right)
$-positive $$R>|A_{11}|.$$ Then $\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$ is a pseudo-Einstein contact form for any CR-pluriharmonic function $f$ if and only if the solvability of the inhomogeneous tangential Cauchy-Riemann equation (\[2018A\]).

We observe that, for a strictly pseudoconvex $3$-manifold $(M^{3},J,\theta)$, we have the invariance property for the CR-pluriharmonic operator $P_{1}$ and CR Paneitz operator $P_{0}$. It is to say that, for rescaled contact form $\widetilde{\theta}=e^{2g}\theta,$ we have $$\widetilde{P}_{1}=e^{-3g}P_{1}\text{ \textrm{and}\  \ }\widetilde{P}_{0}=e^{-4g}P_{0}.\label{A}$$ Then the nonnegativity of CR Paneitz operator $P_{0}$ is CR conformal invariant ([@h]).

Since the CR Paneitz operator $P_{0}$ is nonnegative ([@ccc]) if the pseudohermitian torsion is vanishing, it follows from Theorem \[t2\] and Corollary \[c2\] that

\[c3\] 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 we have the solvability of the inhomogeneous tangential Cauchy-Riemann equation (\[2018A\]). That is to say that the Kohn–Rossi cohomology class of $\sigma_{\overline{1}}\theta^{\overline{1}}$ is vanishing.

Let $(M,J,\theta)$ be a closed strictly pseudoconvex CR $3$-manifold with $c_{1}(T_{1,0}M)=0.$ With the notations as in section $2$ and section $3,$ another Bôchner-type equality holds$$\int_{M}\left(  R-\frac{1}{2}Tor-\frac{1}{2}Tor^{\prime}\right)  \left(
\gamma,\gamma \right)  d\mu+\int_{M}\left \vert \gamma_{1,1}\right \vert ^{2}d\mu+\int_{M}Qud\mu+\int_{M}\left(  P_{0}u\right)  ud\mu=0.$$ With the help of the notion of $\left(  C_{0},C_{1}\right)  $*-convexity*, we have the eigenvalue estimate for the CR Paneitz operator $P_{0}$ in terms of the CR  $Q$-curvature*.*

\[t3\] 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 $$R>(|A_{11}|+\left \vert A_{11,\overline{1}}\right \vert ).$$ If $\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$ is a pseudo-Einstein contact form for any CR-pluriharmonic function $f$, then one can derive the upper bound estimate for the first eigenvalue of the CR Paneitz operator $P_{0}$ $$\Lambda^{2}\int_{M}(u^{\bot})^{2}d\mu \leq \int_{M}(Q^{\bot})^{2}d\mu
\label{2018H}$$ with the decomposition $Q=Q_{\ker}+Q^{\perp}$ and $u=u_{\ker}+u^{\perp}$. Here $\Lambda$ is the positive constant as in (\[41\]).

1\. ([@h]) For a closed strictly pseudoconvex CR $3$-manifold of vanishing pseudohermitian torsion (Sasakian), we have $$\ker P_{1}=\ker P_{0}.\label{2018C}$$ Furthermore, the real ellipsoids in $\mathbf{C}^{2}$ are such that the CR Paneitz operators are nonnegative with kernel consisting of the CR-pluriharmonic functions ([@ccya]). In general for non-embeddable CR $3
$-manifolds, we only have $$\ker P_{1}\varsubsetneq \ker P_{0}.$$

2\. As in the Remark \[r2\], $M$ 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 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 $$R>(|A_{11}|+\left \vert A_{11,\overline{1}}\right \vert )$$ and the CR $Q$-curvature is vanishing $$\Delta_{b}R-i(A_{11,\bar{1}\bar{1}}-A_{\bar{1}\bar{1},11})=0.$$ Then $\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$ is a pseudo-Einstein contact form for any CR-pluriharmonic function $f.$

\[c6\] 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 and the CR $Q$-curvature is vanishing. 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.

We conclude this introduction with a brief plan of the paper. In Section $2$, we derive some preliminary results and indicate the geometry and topology of CR $3$-manifolds with the positivity of pseudohermitian curvature. In Section $3,\ $we prove main Theorems. In Appendix, we survey basic notions in the pseudohermitian (strictly pseudoconvex CR) geometry.

**Acknowledgements** Part of the project was done during the visit to Yau Mathematical Sciences Center, Tsinghua University. The authors would like to express their thanks for the warm hospitality.

Preliminaries
=============

In this section, we derive some necessary ingredients for the proof of main results in this paper. In particular, we define the positivity of pseudohermitian curvature and indicate the geometry and topology of strictly pseudoconvex CR $3$-manifolds. Let $C_{0},C_{1}$ be both nonnegative numbers.

We say that a strictly pseudoconvex CR $3$-manifolds is $\left(  C_{0},C_{1}\right)  $*-convex* if the pseudohermitian curvature is $\left(
C_{0},C_{1}\right)  $-positive. That is$$\begin{array}
[c]{ccl}(R-C_{0}Tor-C_{1}Tor^{\prime})(X,X) & = & Rx^{1}x^{\bar{1}}-2C_{0}\mathrm{Re}[i(A_{\bar{1}\bar{1}}x^{\bar{1}}x^{\bar{1}})]\\
&  & -2C_{1}\mathrm{Re[}(iA_{\bar{1}\bar{1},1}x^{\bar{1}})]>0,
\end{array}
\label{33}$$ $\mathrm{for\ any}\ X=x^{1}Z_{1}\in T_{1,0}(M).$

Before giving the proof of Theorem \[t3\], we explain why we introduce the notion of $\left(  C_{0},C_{1}\right)  $-convexity as follows:

\[l6\]Let $M$ be a closed strictly pseudoconvex CR $3$-manifold. For any nonnegative constant $C_{0},C_{1}$; $\left(  C_{0},C_{1}\right)  $-convexity is equivalent to the curvature-torsion pinching condition $$R(x)>2\left(  C_{0}|A_{11}|+C_{1}\left \vert A_{11,\overline{1}}\right \vert
\right)  (x)$$ for all $x\in M$.

\[r2\] ([@cacc]) Let $(M^{3},J,\theta)$ be a closed strictly pseudoconvex CR $3$-manifold with $$R(x)>\left \vert A_{11}(x)\right \vert$$ which is $\left(  C_{0},0\right)  $-positive. Then $M$ admits a Riemannian metric of positive scalar curvature.

Fix a point $x\in M$ and denote $$\left \{
\begin{array}
[c]{l}A_{11}(x)=a(x)+b(x)i\\
A_{11,\overline{1}}(x)=c(x)+d(x)i
\end{array}
\right.  .$$ Without loss of generality, one may consider $X=x^{1}Z_{1}=(1+si)Z_{1}$ for some $s\in \mathbb{R}$. The convexity condition $\left(  \ref{33}\right)  $ reads as $$\begin{array}
[c]{ccl}R(1+si)(1-si) & > & i[C_{0}(a-bi)(1-si)^{2}+C_{1}\left(  c-di\right)
(1-si)-conj.]\\
& = & -2\left[  \left(  C_{0}b\right)  s^{2}-\left(  2C_{0}a+C_{1}c\right)
s+\left(  -C_{1}d-C_{0}b\right)  \right]  .
\end{array}$$ i.e., $$R>f\left(  s\right)  :=-2\left[  \left(  C_{0}b\right)  -\frac{\left(
2C_{0}b+C_{1}d\right)  +\left(  2C_{0}a+C_{1}c\right)  s}{1+s^{2}}\right]
.\label{41b}$$ Since $X$ is arbitrary, this inequality holds for all $s\in \mathbb{R}$. We have $$\begin{array}
[c]{ccl}f(s) & \leq & \max_{s\in \mathbb{R}}f(s)\\
& = & f(s_{0})\\
& = & 2\sqrt{\left(  C_{0}a+C_{1}\frac{c}{2}\right)  ^{2}+\left(  C_{0}b+C_{1}\frac{d}{2}\right)  ^{2}}+C_{1}d\\
& \leq & 2\left(  C_{0}|A_{11}|+C_{1}\left \vert A_{11,\overline{1}}\right \vert
\right)
\end{array}
\label{41a}$$ where $$s_{0}=\frac{-\left(  C_{0}b+C_{1}\frac{d}{2}\right)  +\sqrt{\left(
C_{0}a+C_{1}\frac{c}{2}\right)  ^{2}+\left(  C_{0}b+C_{1}\frac{d}{2}\right)
^{2}}}{\left(  C_{0}a+C_{1}\frac{c}{2}\right)  },$$ for $C_{0}a+C_{1}\frac{c}{2}\neq0$, is a critical number of $f$. Thus $$R(x)>2\left(  C_{0}|A_{11}|+C_{1}\left \vert A_{11,\overline{1}}\right \vert
\right)  (x)$$ for all $x\in M$. As for the case of $C_{0}a+C_{1}\frac{c}{2}=0$, the same deduction could be applied. Therefore, $\left(  C_{0},C_{1}\right)
$-positivity is equivalent to $$R>2\left(  C_{0}|A_{11}|+C_{1}\left \vert A_{11,\overline{1}}\right \vert
\right)  .$$

([@l]) (i) A contact form $\theta$ on a closed strictly pseudoconvex CR $(2n+1)$-manifold $(M,\theta)$ is said to be pseudo-Einstein for $n\geq2$ if the pseudohermitian Ricci tensor $R_{\alpha \overline{\beta}}$ is proportional to the Levi form $h_{\alpha \overline{\beta}}$, i.e., $$\ R_{\alpha \overline{\beta}}=\frac{R}{n}h_{\alpha \overline{\beta}},$$ where $R=h^{\alpha \overline{\beta}}R_{\alpha \overline{\beta}}$ is the Tanaka-Webster scalar curvature of $(J,\theta).$

\(ii) (Lemma \[l1\]) Note that any contact form on a closed strictly pseudoconvex $3$-manifold is actually pseudo-Einstein (since the pseudohermitian Ricci tensor has only one component $R_{1\overline{1}}$). Then we define a contact form $\theta$ on a closed strictly pseudoconvex CR $3$-manifold $(M,\theta)$ is said to be pseudo-Einstein if the following tensor is vanishing $$W_{1}\doteqdot \left(  R_{1}-iA_{11_{,}\overline{1}}\right)  =0.$$ (iii) We define the first Chern class $c_{1}(T_{1,0}M)\in H^{2}(M,\mathbf{R})
$ for the holomorphic tangent bundle $T^{1,0}M$ by $$\begin{aligned}
c_{1}(T^{1,0}M)  & =\frac{i}{2\pi}[d\omega_{\alpha}{}^{\alpha}]\\
& =\frac{i}{2\pi}[R_{\alpha \overline{\beta}}\theta^{\alpha}\wedge
\theta^{\overline{\beta}}+A_{\alpha \mu,\overline{\alpha}}{}\theta^{\mu}\wedge \theta-A_{\overline{\alpha}\overline{\mu},\alpha}\theta^{\overline{\mu}}\wedge \theta].\end{aligned}$$ (iv) Note that any pseudo-Einstein manifold $(M^{2n+1},\theta)$, the first Chern class $c_{1}(T_{1,0}M)$ of $T_{1,0}(M)$ is vanishing ([@l]).

Next let us recall the equivalent definitions of the pseudo-Einsteinian $(2n+1)$-manifold for $n\geq2$ and $n=1$ as well.

\[l1\] (i) If $(M,J,\theta)$ is a strictly pseudoconvex CR $(2n+1)$-manifold for $n\geq2$, then the following propositions are all equivalent :$$\begin{array}
[c]{cl}\left(  1\right)  & R_{\alpha \overline{\beta}}=\frac{R}{n}h_{\alpha
\overline{\beta}},\text{ \  \ }\\
\left(  2\right)  & \left(  \omega_{\alpha}^{\alpha}+\frac{i}{n}R\theta \right)  \text{ }\mathrm{is}\text{ }\mathrm{closed,}\\
\left(  3\right)  & W_{\alpha}\doteqdot \left(  R,_{\alpha}-inA_{\alpha \beta
},^{\beta}\right)  =0\text{.\ }\end{array}$$ As for $n=1,$ we still have the equivalent between $(2)$ and $(3).$

\(ii) By the equivalence of $\left(  2\right)  $ and $\left(  3\right)  $, we see the first Chern class $c_{1}(T_{1,0}M)$ is vanishing if $(M,J,\theta)$ is a pseudo-Einsteinian $3$-manifold.

The equivalence of $\left(  1\right)  $ and $\left(  2\right)  $ could be found in [@l] for $n\geq2$. The proof of $\left(  2\right)
\Longleftrightarrow \left(  3\right)  $ for $n\geq2$ is the same with $n=1$. So, for simplification, we just give the proof of the equivalence of $\left(
2\right)  \Longleftrightarrow \left(  3\right)  $ for $n=1$.

Because $$d\omega_{1}^{1}=R\theta^{1}\wedge \theta^{\overline{1}}+A_{11,\overline{1}}\theta^{1}\wedge \theta-A_{\overline{1}\overline{1},1}\theta^{\overline{1}}\wedge \theta,$$ we have$$\begin{array}
[c]{ccl}d\left(  \omega_{1}^{1}+iR\theta \right)  & = & d\omega_{1}^{1}+i\left(
R_{,1}\theta^{1}+R_{,\overline{1}}\theta^{\overline{1}}\right)  \wedge
\theta-R\theta^{1}\wedge \theta^{\overline{1}}\\
& = & i\left[  \left(  R_{,1}-iA_{11,\overline{1}}\right)  \theta^{1}+\left(
R_{,\overline{1}}+iA_{\overline{1}\overline{1},1}\right)  \theta^{\overline
{1}}\right]  \wedge \theta
\end{array}
.$$

Hence $$d\left(  \omega_{1}^{1}+iR\theta \right)  =0\Longleftrightarrow R_{,1}-iA_{11,\overline{1}}=0.$$

We recall some useful notations as well.

([@l]) (i) Let $(M,J,\theta)$ be a three-dimensional strictly pseudoconvex CR manifold. We define $$P\varphi=(P_{1}\varphi)\theta^{1},$$ which is an operator that characterizes CR-pluriharmonic functions. Here $P_{1}\varphi=\varphi_{\bar{1}}{}^{\bar{1}}{}_{1}+iA_{11}\varphi^{1}$ and $\overline{P}\varphi=(\overline{P}_{1})\theta^{\bar{1}}$, the conjugate of $P$. The CR Paneitz operator $P_{0}$ is defined by$$P_{0}\varphi=\left(  \delta_{b}(P\varphi)+\overline{\delta}_{b}(\overline
{P}\varphi)\right)  ,\label{id9}$$ where $\delta_{b}$ is the divergence operator that takes $(1,0)$-forms to functions by $\delta_{b}(\sigma_{1}\theta^{1})=\sigma_{1,}{}^{1}$ and, similarly, $\bar{\delta}_{b}(\sigma_{\bar{1}}\theta^{\bar{1}})=\sigma_{\bar
{1},}{}^{\bar{1}}$. We observe that$$\int_{M}\langle P\varphi+\overline{P}\varphi,d_{b}\varphi \rangle_{L_{\theta
}^{\ast}}\ d\mu=-\int_{M}P_{0}\varphi \cdot \varphi \ d\mu \label{10}$$ with $d\mu=\theta \wedge d\theta.$ One can check that $P_{0}$ is self-adjoint, that is, $\left \langle P_{0}\varphi,\psi \right \rangle =\left \langle
\varphi,P_{0}\psi \right \rangle $ for all smooth functions $\varphi$ and $\psi
$. For the details about these operators, the reader can make reference to [@gl], [@h], [@l], [@gg] and [@fh].

\(ii) On a complete pseudohermitian $3$-manifold $(M,J,\theta),$ we call the Paneitz operator $P_{0}$ with respect to $(J,\theta)$ essentially positive if there exists a constant $\Lambda$ $>$ $0$ such that $$\int_{M}P_{0}\varphi \cdot \varphi d\mu \geq \Lambda \int_{M}\varphi^{2}d\mu.\label{41}$$ for all real smooth functions $\varphi$ $\in(\ker P_{0})^{\perp}$ (i.e. perpendicular to the kernel of $P_{0}$ in the $L^{2}$ norm with respect to the volume form $d\mu$ $=$ $\theta \wedge d\theta).$ We say that $P_{0}$ is nonnegative if $$\int_{M}P_{0}\varphi \cdot \varphi d\mu \geq0$$ for all real smooth functions $\varphi$.

\[r1\] 1. The notions of Paneitz operator $P_{0}$ and $Q$-curvature were initially introduced on a Riemannian manifold, and were considered as a kind of generalization of Laplacian and Gaussian curvature on a two-dimensional manifold, respectively ([@h]).

2\. The kernel of the CR Paneitz operator $P_{0}$ is infinite dimensional, containing all CR-pluriharmonic functions.

3\. Let $(M,J,\theta)$ be a closed strictly pseudoconvex $3$-manifold with vanishing pseudohermitian torsion. Then the corresponding CR Paneitz operator $P_{0}$ is essentially positive ( [@ccc], [@cac]).

Finally, we define the CR $Q$-curvature in a pseudohermitian $3$-manifold by $$Q:=-\mathrm{Re}(R_{_{,}1}-iA_{11,\bar{1}})_{\bar{1}}=-\mathrm{Re}(R_{,1\bar
{1}}-iA_{11,\bar{1}\bar{1}}).\label{c}$$ Then $$Q=-\frac{1}{2}[\Delta_{b}R-i(A_{11,\bar{1}\bar{1}}-A_{\bar{1}\bar{1},11})].$$ Now for $\theta=e^{2\gamma}\theta_{0}$, under this conformal change, it is known that we have the following transformation laws ([@h]) : $$Q=e^{-4\gamma}(Q_{0}+\frac{3}{4}P_{0}\gamma)\label{0b}$$ and $$W_{1}:=(R_{,1}-iA_{11},_{\bar{1}})=e^{-3\gamma}[\overset{0}{R}_{,1}-i\overset{0}{A}_{11,\bar{1}}-6\overset{0}{P}_{1}\gamma],\label{0c}$$ where $P_{0}$ and $Q_{0}$ denote the CR Paneitz operator and the CR $Q$-curvature with respect to $(M,J,$ $\theta_{0})$, respectively.

Finally, we recall that

We call a CR structure $J$ spherical if Cartan curvature tensor $Q_{11}$ vanishes identically. Here $$Q_{11}=\frac{1}{6}R_{11}+\frac{i}{2}RA_{11}-A_{11,0}-\frac{2i}{3}A_{11,\overset{\_}{1}1}.$$ Note that $(M,J,\theta)$ is called a spherical CR $3$-manifold if $J$ is a spherical structure. We observe that the spherical structure is CR invariant and a closed spherical CR $3$-manifold $(M,J,\theta)$ is locally CR equivalent to the standard CR $3$-sphere $(\mathbf{S}^{3},\widehat{J},\widehat{\theta}).$ In additional, if $M$ is simply connected, then $(M,J,\theta)$ is the standard CR $3$-sphere.

Proofs of Main Theorems
=======================

In this section, we prove the main theorems. We start from the groundwork for Theorem \[t1\].

\[l2\] If $(M,J,\theta)$ is a strictly pseudoconvex $3$-manifold with $c_{1}(T_{1,0}M)=0$, then there is a pure imaginary $1$-form $$\sigma=\sigma_{\overline{1}}\theta^{\overline{1}}-\sigma_{1}\theta^{1}+i\sigma_{0}\theta$$ with $d\omega_{1}^{1}=d\sigma$ such that$$\left \{
\begin{array}
[c]{l}R=R_{1\overline{1}}=\sigma_{\overline{1},1}+\sigma_{1,\overline{1}}-\sigma
_{0}\\
A_{11,\overline{1}}=\sigma_{1,0}+i\sigma_{0,1}-A_{11}\sigma_{\overline{1}}\end{array}
\right.  .\label{1}$$

Because$$c_{1}(T_{1,0}M)=-\frac{1}{2\pi i}\left[  d\omega_{1}^{1}\right]  =0,$$

we know there is a pure imaginary $1$-form$$\sigma=\sigma_{\overline{1}}\theta^{\overline{1}}-\sigma_{1}\theta^{1}+i\sigma_{0}\theta$$ such that$$d\omega_{1}^{1}=d\sigma.$$

By the structure equation$$\left \{
\begin{array}
[c]{l}d\theta=i\theta^{1}\wedge \theta^{\overline{1}}\\
d\theta^{1}=A_{\overline{1}\overline{1}}\theta \wedge \theta^{1}\end{array}
\right.  ,$$ we have$$\begin{array}
[c]{ccl}d\sigma & = & \left(  \sigma_{\overline{1},1}\theta^{1}+\sigma_{\overline
{1},0}\theta \right)  \wedge \theta^{\overline{1}}+\sigma_{\overline{1}}d\theta^{\overline{1}}-\left(  \sigma_{1,\overline{1}}\theta^{\overline{1}}+\sigma_{1,0}\theta \right)  \wedge \theta^{1}-\\
&  & \sigma_{1}d\theta^{1}+i\left(  \sigma_{0,1}\theta^{1}+\sigma
_{0,\overline{1}}\theta^{\overline{1}}\right)  \wedge \theta+i\sigma_{0}d\theta \\
& = & \left(  \sigma_{\overline{1},1}+\sigma_{1,\overline{1}}-\sigma
_{0}\right)  \theta^{1}\wedge \theta^{\overline{1}}-\left(  \sigma
_{1,0}+i\sigma_{0,1}-\sigma_{\overline{1}}A_{11}\right)  \theta \wedge
\theta^{1}+\\
&  & \left(  \sigma_{\overline{1},0}-i\sigma_{0,\overline{1}}-\sigma
_{1}A_{\overline{1}\overline{1}}\right)  \theta \wedge \theta \overline{^{1}}.
\end{array}
.$$

Due to $$d\sigma=d\omega_{1}^{1}=R_{1\overline{1}}\theta^{1}\wedge \theta^{\overline{1}}+A_{11,\overline{1}}\theta^{1}\wedge \theta-A_{\overline{1}\overline{1},1}\theta^{\overline{1}}\wedge \theta,$$ we derive$$\left \{
\begin{array}
[c]{l}R=R_{1\overline{1}}=\sigma_{\overline{1},1}+\sigma_{1,\overline{1}}-\sigma
_{0}\\
A_{11,\overline{1}}=\sigma_{1,0}+i\sigma_{0,1}-A_{11}\sigma_{\overline{1}}\end{array}
\right.  .$$

We would need the J.J. Kohn’s Hodge theory for the $\overline{\partial}_{b}$ complex (see [@k]) :

\[l3\]If $(M,J,\theta)$ is an embeddable closed strictly pseudoconvex CR $3$-manifold and $\eta \in \Omega^{0,1}\left(  M\right)  $, a smooth $\left(
0,1\right)  $-form on $M$ with $$\overline{\partial}_{b}\eta=0,$$ then there are a complex function $\varphi \in C_{\mathbb{C}
}^{\infty}\left(  M\right)  $ and $\gamma \in \Omega^{0,1}\left(  M\right)  $ such that $$\left(  \eta-\overline{\partial}_{b}\varphi \right)  =\gamma \in \ker \left(
\square_{b}\right)  ,$$ where $\square_{b}=2\left(  \overline{\partial}_{b}\overline{\partial}_{b}^{\ast}+\overline{\partial}_{b}^{\ast}\overline{\partial}_{b}\right)  $ is the Kohn-Rossi Laplacian.

Subsequently, we deduce the expression for $W_{1}$. We denote $\gamma
_{1}:=\overline{\gamma_{\overline{1}}}.$

\[l4\] If $(M,J,\theta)$ is an embeddable closed strictly pseudoconvex CR $3$-manifold with $c_{1}(T_{1,0}M)=0$, then there are $u\in C_{\mathbb{R}
}^{\infty}\left(  M\right)  $ and $\gamma=\gamma_{\overline{1}}\theta
^{\overline{1}}\in \Omega^{0,1}\left(  M\right)  $ with $$\gamma_{\overline{1},1}=\gamma_{1,\overline{1}}=0$$ such that $$W_{1}=2P_{1}u+i\left(  A_{11}\gamma_{\overline{1}}-\gamma_{1,0}\right)
.\label{0}$$

By choosing $\eta=\sigma_{\overline{1}}\theta^{\overline{1}}$ as in Lemma \[l3\], where $\sigma$ is chosen from Lemma \[l2\],** **there are$$\varphi=u+iv\in C_{\mathbb{C}
}^{\infty}\left(  M\right)$$ and$$\gamma=\gamma_{\overline{1}}\theta^{\overline{1}}\in \Omega^{0,1}\left(
M\right)  \cap \ker \left(  \square_{b}\right)$$ such that$$\sigma_{\overline{1}}=\varphi_{\overline{1}}+\gamma_{\overline{1}}\label{18}$$

Note that$$\square_{b}\gamma=0\Longrightarrow \overline{\partial}_{b}^{\ast}\gamma=0\Longrightarrow \gamma_{\overline{1},1}=0\label{2}$$ and$$\sigma_{1}=\left(  \overline{\varphi}\right)  _{1}+\gamma_{1}.\label{19}$$

Thus$$\begin{array}
[c]{ccl}\sigma_{1,\overline{1}1} & = & (\overline{\varphi})_{,1\overline{1}1}+\gamma_{1,\overline{1}1}\text{ \ }\mathrm{by}\text{ }\left(  \ref{19}\right)
\\
& = & (\overline{\varphi})_{,1\overline{1}1}\text{ \ }\mathrm{by}\text{
}\left(  \ref{2}\right) \\
& = & (\overline{\varphi})_{,\overline{1}11}+i(\overline{\varphi})_{,01}\text{
\ }\mathrm{by}\text{ }\left(  \ref{6}\right) \\
& = & (\overline{\varphi})_{,\overline{1}11}+i\left[  (\overline{\varphi
})_{,10}+A_{11}(\overline{\varphi})_{,\overline{1}}\right]  \text{
\ }\mathrm{by}\text{ }\left(  \ref{6}\right)
\end{array}$$ and$$\sigma_{\overline{1},11}=\varphi_{,\overline{1}11}\text{ \ }\mathrm{from}\text{ }\left(  \ref{18}\right)  \text{ }\mathrm{and}\text{ }\left(
\ref{2}\right)$$ imply$$\begin{array}
[c]{ccl}W_{1} & = & R_{,1}-iA_{11,\overline{1}}\\
& = & \sigma_{\overline{1},11}+\sigma_{1,\overline{1}1}-i\sigma_{1,0}+iA_{11}\sigma_{\overline{1}}\text{ \ }\mathrm{by}\text{ }\left(
\ref{1}\right) \\
& = & \varphi_{,\overline{1}11}+(\overline{\varphi})_{,\overline{1}11}+iA_{11}(\overline{\varphi})_{,\overline{1}}-i\gamma_{1,0}+iA_{11}\left(
\varphi_{\overline{1}}+\gamma_{\overline{1}}\right) \\
& = & 2\left(  u_{\overline{,1}11}+iA_{11}u_{\overline{1}}\right)  +i\left(
A_{11}\gamma_{\overline{1}}-\gamma_{1,0}\right) \\
& = & 2P_{1}u+i\left(  A_{11}\gamma_{\overline{1}}-\gamma_{1,0}\right)
\end{array}
.$$

This completes the proof.

Now we are ready to give the proof of Theorem \[t1\]:

(Proof of Theorem \[t1\]**)** Set$$\widetilde{\theta}=e^{2\lambda}\theta \text{.}$$ By the transformation law (refer to Lemma 5.4 in [@h] or Lemma 3.1 in [@cw]), we know$$\widetilde{W_{1}}=e^{-3\lambda}\left(  W_{1}-6P_{1}\lambda \right)  ,\label{5}$$ where the notation with “tilde” means such quantity corresponds to the new contact form $\widetilde{\theta}$. With the help of $\left(  \ref{5}\right)  $ and Lemma \[l4\], we have$$\widetilde{W_{1}}=0$$ if and only if$$W_{1}=6P_{1}\lambda$$ if and only if $$6P_{1}\lambda=2P_{1}u+i\left(  A_{11}\gamma_{\overline{1}}-\gamma
_{1,0}\right)  .$$ That is to say $$P_{1}f=i\left(  A_{11}\gamma_{\overline{1}}-\gamma_{1,0}\right)$$ for $$f=\left(  6\lambda-2u\right)  .$$

From $\left(  \ref{7}\right)  $ and $$\gamma_{1,0\overline{1}}=\gamma_{1,\overline{1}0}+A_{\overline{1}\overline{1}}\gamma_{1,1}+A_{\overline{1}\overline{1},1}\gamma_{1}=A_{\overline
{1}\overline{1}}\gamma_{1,1}+A_{\overline{1}\overline{1},1}\gamma_{1},$$ we could deduce $f$ satisfies the fourth-order partial differential equation $$P_{0}f=2i\left[  \left(  A_{11}\gamma_{\overline{1}}\right)  _{,\overline{1}}-\left(  A_{\overline{1}\overline{1}}\gamma_{1}\right)  _{,1}\right]
\label{24}$$ where $P_{0}$ is the CR Paneitz  operator (see section$\ 2$). This suggests us there is an obstruction to the existence of pseudo-Einstein contact form pertaining to the CR Paneitz operator. See Theorem \[t2\] below for more details.

As for the proof of the case of vanishing pseudohermitian torsion :

(Proof of Corollary \[c2\])

Setting $A_{11}=0$ in $\left(  \ref{13}\right)  $, by Theorem \[t1\], it suffices to show that $$\gamma_{1,0}=0$$ in order to have a globally defined pseudo-Einstein contact form $\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$.

Note that, from $\left(  \ref{0}\right)  $ and $A_{11}=0$, $$R_{,1}=2u_{\overline{1}11}-i\gamma_{1,0}.\label{8}$$

Utilizing integration by parts, it follows from (\[8\]) and $\gamma
_{\overline{1},1}=0$ that $$\begin{array}
[c]{ccl}0 & \leq & \int_{M}\left \vert \gamma_{1,0}\right \vert ^{2}d\mu \\
& = & -\int_{M}\gamma_{1}\gamma_{\overline{1},00}d\mu \\
& = & -\int_{M}\gamma_{1}\left(  -iR_{,\overline{1}}+2iu_{,1\overline
{1}\overline{1}}\right)  _{,0}d\mu \  \\
& = & i\int_{M}\gamma_{1}\left(  R_{,0}-2u_{,1\overline{1}0}\right)
_{,\overline{1}}d\mu \\
& = & -i\int_{M}\gamma_{1,\overline{1}}\left(  R_{,0}-2u_{,1\overline{1}0}\right)  d\mu \\
& = & 0.
\end{array}
.$$

The third equality comes from $\left(  \ref{6}\right)  $ and $A_{11}=0$. Then $$\gamma_{1,0}=0.$$

Before giving the proof of Theorem \[t2\], we need the following Bôchner-type equality.

\[l8\] Let $(M,J,\theta)$ be a closed strictly pseudoconvex CR $3$-manifold and $\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$ is a pseudo-Einstein contact form. Then we have$${\displaystyle \int \limits_{M}}
\left(  2R-Tor\right)  \left(  \gamma,\gamma \right)  d\mu+2{\displaystyle \int \limits_{M}}
\left \vert \gamma_{1,1}\right \vert ^{2}d\mu+\frac{1}{2}{\displaystyle \int \limits_{M}}
\left(  P_{0}f\right)  fd\mu=0.\label{26A}$$

From Theorem \[t1\] and the commutation formula, it follows that$$\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$$ is a pseudo-Einstein contact form if and only if $$P_{1}f=iA_{11}\gamma_{\overline{1}}+R\gamma_{1}-\gamma_{1,1\overline{1}}.\label{27A}$$

By the fact that $\gamma_{1,\overline{1}}=0$, it’s easy to see $$\int_{M}\left(  P_{1}f\right)  \gamma_{\overline{1}}d\mu=\int_{M}\left(
f_{\overline{1}11}+iA_{11}f_{\overline{1}}\right)  \gamma_{\overline{1}}d\mu=i\int_{M}A_{11}f_{\overline{1}}\gamma_{\overline{1}}d\mu.$$

Then, substituting $\left(  \ref{27A}\right)  $ into the last equality and adding its conjugation, we have$$-\int_{M}Tor\left(  d_{b}f,\gamma \right)  d\mu=\int_{M}\left(  2R-Tor\right)
\left(  \gamma,\gamma \right)  d\mu+2\int_{M}\left \vert \gamma_{1,1}\right \vert
^{2}d\mu.\label{28A}$$

On the other hand, the equality $\left(  \ref{27A}\right)  $ and the commutation formulas enable us to get$$\begin{array}
[c]{ccl}\int_{M}\left(  P_{1}f\right)  f_{\overline{1}}d\mu & = & \int_{M}\left(
iA_{11}\gamma_{\overline{1}}+R\gamma_{1}\right)  f_{\overline{1}}d\mu-\int
_{M}\gamma_{1}f_{\overline{1}\overline{1}1}d\mu \\
& = & \int_{M}\left(  iA_{11}\gamma_{\overline{1}}+R\gamma_{1}\right)
f_{\overline{1}}d\mu \\
&  & +\int_{M}\gamma_{1}\left(  -f_{\overline{1}1\overline{1}}+if_{\overline
{1}0}-R\gamma_{1}f_{\overline{1}}\right)  d\mu \\
& = & \int_{M}iA_{11}\gamma_{\overline{1}}f_{\overline{1}}d\mu+\int_{M}\gamma_{1}\left(  -f_{\overline{1}1\overline{1}}+if_{\overline{1}0}\right)
d\mu \\
& = & \int_{M}iA_{11}\gamma_{\overline{1}}f_{\overline{1}}d\mu+\int_{M}\gamma_{1}\left(  -f_{\overline{1}1\overline{1}}+if_{0\overline{1}}-iA_{\overline{1}\overline{1}}f_{1}\right)  d\mu \\
& = & i\int_{M}\left(  A_{11}\gamma_{\overline{1}}f_{\overline{1}}-A_{\overline{1}\overline{1}}\gamma_{1}f_{1}\right)  d\mu \\
& = & -\int_{M}Tor\left(  d_{b}f,\gamma \right)  d\mu.
\end{array}$$

By the definition of  the CR Paneitz operator, we obtain $$\int_{M}\left(  P_{0}f\right)  fd\mu=-\int_{M}\left(  \left(  P_{1}f\right)
f_{\overline{1}}+\left(  P_{\overline{1}}f\right)  f_{1}\right)  d\mu
=2\int_{M}Tor\left(  d_{b}f,\gamma \right)  d\mu \label{29A}$$

Therefore, it follows from the equalities $\left(  \ref{28A}\right)  $ and $\left(  \ref{29A}\right)  $ that $$\int_{M}\left(  2R-Tor\right)  \left(  \gamma,\gamma \right)  d\mu+2\int
_{M}\left \vert \gamma_{1,1}\right \vert ^{2}d\mu+\frac{1}{2}\int_{M}\left(
P_{0}f\right)  fd\mu=0.$$

Then we are done.

Such equality enables us to prove Theorem \[t2\] as follows:

(Proof of Theorem \[t2\]) From the equality $\left(  \ref{26A}\right)  $ and the hypotheses, it is clear that if $\widetilde{\theta}=e^{\frac{\left(
f+2u\right)  }{3}}\theta$ is a pseudo-Einstein contact form, then$$\gamma=0.$$ Hence we can solve the inhomogeneous tangential Cauchy-Riemann equation $$\overline{\partial}_{b}\varphi=\sigma_{\overline{1}}\theta^{\overline{1}}$$ by Lemma \[l3\]. Note that this implicitly implies $f$ is CR-pluriharmonic. So the sufficient part is completed.

As for the necessary part, it’s obvious from Theorem \[t1\].

Before to go further, we need the following key lemma.

\[l5\] Let $(M,J,\theta)$ be a closed strictly pseudoconvex CR $3$-manifold with $c_{1}(T_{1,0}M)=0.$ With the notations as above, the following equality holds$$\int_{M}\left(  R-\frac{1}{2}Tor-\frac{1}{2}Tor^{\prime}\right)  \left(
\gamma,\gamma \right)  d\mu+\int_{M}\left \vert \gamma_{1,1}\right \vert ^{2}d\mu+\int_{M}Qud\mu+\int_{M}\left(  P_{0}u^{\bot}\right)  u^{\bot}d\mu=0.\label{2018BB}$$

From the equality $\left(  \ref{0}\right)  $, we are able to get$$\begin{array}
[c]{ccl}\left(  R_{,1}-iA_{11,\overline{1}}\right)  \gamma_{\overline{1}} & = &
W_{1}\gamma_{\overline{1}}\\
& = & 2\left(  u_{\overline{1}11}+iA_{11}u_{\overline{1}}\right)
\gamma_{\overline{1}}+iA_{11}\gamma_{\overline{1}}\gamma_{\overline{1}}-i\gamma_{1,0}\gamma_{\overline{1}}\\
& = & 2\left(  u_{\overline{1}11}+iA_{11}u_{\overline{1}}\right)
\gamma_{\overline{1}}+iA_{11}\gamma_{\overline{1}}\gamma_{\overline{1}}-\left(  \gamma_{1,1\overline{1}}-R\gamma_{1}\right)  \gamma_{\overline{1}}.
\end{array}$$

Taking the integration over $M$ of both sides and its conjugation, we have, by the fact that $\gamma_{1,\overline{1}}=0$, $$i\int_{M}\left(  A_{11,\overline{1}}\gamma_{\overline{1}}-A_{\overline
{1}\overline{1},1}\gamma_{1}\right)  d\mu+\int_{M}\left(  2R-Tor\right)
\left(  \gamma,\gamma \right)  d\mu+2\int_{M}\left \vert \gamma_{1,1}\right \vert
^{2}d\mu-2\int_{M}Tor\left(  d_{b}u,\gamma \right)  d\mu=0.$$ That is $$\int_{M}\left(  R-\frac{1}{2}Tor-\frac{1}{2}Tor^{\prime}\right)  \left(
\gamma,\gamma \right)  d\mu+\int_{M}\left \vert \gamma_{1,1}\right \vert ^{2}d\mu-\int_{M}Tor\left(  d_{b}u,\gamma \right)  d\mu=0.\label{34}$$ On the other hand, it follows from the equality $\left(  \ref{0}\right)  $ that $$\left(  R,_{1}-iA_{11},_{\overline{1}}\right)  u_{\overline{1}}=W_{1}u_{\overline{1}}=\left[  2P_{1}u+i\left(  A_{11}\gamma_{\overline{1}}-\gamma_{1,0}\right)  \right]  u_{\overline{1}}.\label{30A}$$ By the fact that $\gamma_{1,\overline{1}}=0$, we see that$$\begin{array}
[c]{ccl}\int_{M}\gamma_{1,0}u_{\overline{1}}d\mu & = & \int_{M}\gamma_{1}u_{\overline{1}0}d\mu \\
& = & -\int_{M}\gamma_{1}\left(  u_{0\overline{1}}-A_{\overline{1}\overline
{1}}u_{1}\right)  d\mu \\
& = & \int_{M}A_{\overline{1}\overline{1}}u_{1}\gamma_{1}d\mu.
\end{array}
\label{31A}$$ It follows from $\left(  \ref{30A}\right)  \ $and$\  \left(  \ref{31A}\right)
$ that$$\begin{array}
[c]{l}\  \  \ 2\int_{M}Qud\mu+2\int_{M}\left(  P_{0}u\right)  ud\mu \\
=i\int_{M}\left[  \left(  A_{11}u_{\overline{1}}\gamma_{\overline{1}}-A_{\overline{1}\overline{1}}u_{1}\gamma_{1}\right)  -conj\right]  d\mu \\
=-2\int_{M}Tor\left(  d_{b}u,\gamma \right)  d\mu.
\end{array}$$ Thus by (\[34\]) $$\int_{M}\left(  R-\frac{1}{2}Tor-\frac{1}{2}Tor^{\prime}\right)  \left(
\gamma,\gamma \right)  d\mu+\int_{M}\left \vert \gamma_{1,1}\right \vert ^{2}d\mu+\int_{M}Qud\mu+\int_{M}\left(  P_{0}u^{\bot}\right)  u^{\bot}d\mu=0.$$

(proof of Theorem \[t3\] and Corollary \[c4\])  If we assume that $$\ker P_{1}=\ker P_{0}.$$ Then we also have $$0=\int_{M}\left(  R-\frac{1}{2}Tor-\frac{1}{2}Tor^{\prime}\right)  \left(
\gamma,\gamma \right)  d\mu+\int_{M}\left \vert \gamma_{1,1}\right \vert ^{2}d\mu+\int_{M}Qu^{\bot}d\mu+\int_{M}\left(  P_{0}u^{\bot}\right)  u^{\bot}d\mu.\label{2018F}$$ Here we have used the fact that $P_{0}$ is self-adjoint and$$\int_{M}Tor\left(  d_{b}u_{\ker},\gamma \right)  d\mu=0.$$

Now if $\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$ is a pseudo-Einstein contact form for any CR-pluriharmonic function $f,$ it follows from (\[26A\]) that $$\gamma=0$$ and then $$0=\int_{M}Qu^{\bot}d\mu+\int_{M}\left(  P_{0}u^{\bot}\right)  u^{\bot}d\mu=\int_{M}Q^{\bot}u^{\bot}d\mu+\int_{M}\left(  P_{0}u^{\bot}\right)
u^{\bot}d\mu.$$ By  the Hőrder inequality and essentially positivity of the CR Paneitz operator, we have$$\begin{array}
[c]{ccl}\int_{M}Q^{\bot}u^{\bot}d\mu+\int_{M}\left(  P_{0}u^{\bot}\right)  u^{\bot
}d\mu & \geq & \Lambda \int_{M}(u^{\bot})^{2}d\mu-(\int_{M}(Q^{\bot})^{2}d\mu)^{\frac{1}{2}})(\int_{M}(u^{\bot})^{2}d\mu)^{\frac{1}{2}}\\
& \geq & [\Lambda(\int_{M}(u^{\bot})^{2}d\mu)^{\frac{1}{2}}-(\int_{M}(Q^{\bot
})^{2}d\mu)^{\frac{1}{2}}](\int_{M}(u^{\bot})^{2}d\mu)^{\frac{1}{2}}\text{ }\end{array}$$ and then $$0\geq \Lambda(\int_{M}(u^{\bot})^{2}d\mu)^{\frac{1}{2}}-(\int_{M}(Q^{\bot})^{2}d\mu)^{\frac{1}{2}}.$$ Hence $$\int_{M}(Q^{\bot})^{2}d\mu \geq \Lambda^{2}\int_{M}(u^{\bot})^{2}d\mu.$$

Furthermore, if he CR  $Q$-curvature is pluriharmonic (i.e. $Q^{\bot}=0),$ then $$u^{\bot}=0$$ and by (\[0\]) $$W_{1}=0.$$ Hence $\theta$ is also a globally defined pseudo-Einstein contact form. Moreover, if the torsion is vanishing, then $(M,J,\theta)$ is the Sasakian space form.

(proof of Theorem \[t3a\] and Corollary \[c5\]) As before $$\begin{array}
[c]{ccl}\int_{M}Qu^{\bot}d\mu+\int_{M}\left(  P_{0}u^{\bot}\right)  u^{\bot}d\mu & = &
\int_{M}Q^{\bot}u^{\bot}d\mu+\int_{M}\left(  P_{0}u^{\bot}\right)  u^{\bot
}d\mu \\
& \geq & \Lambda \int_{M}(u^{\bot})^{2}d\mu \text{ }\\
& \geq & 0
\end{array}$$ if $$Q^{\bot}=0.\label{2018G}$$ It follows from (\[2018F\]) that $$0\geq \int_{M}\left(  R-\frac{1}{2}Tor-\frac{1}{2}Tor^{\prime}\right)  \left(
\gamma,\gamma \right)  d\mu+\int_{M}\left \vert \gamma_{1,1}\right \vert ^{2}d\mu.\label{2018BBB}$$ if (\[2018G\]) holds. Hence$$\gamma=0\text{ }$$ if the pseudohermitian curavture is $\left(  \frac{1}{2},\frac{1}{2}\right)
$*-positive.* It follows from Theorem \[t2\] that $M$ admits a globally defined pseudo-Einstein contact form $\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$.  

Furthermore, if he CR  $Q$-curvature is pluriharmonic (i.e. $Q^{\bot}=0),$ then $$u^{\bot}=0$$ and by (\[0\]) $$W_{1}=0.$$ Hence $\theta$ is also a globally defined pseudo-Einstein contact form.

Now if $(M,J,\theta)$ is spherical and pseudo-Einstein, we have $$W_{1}=R,_{1}-iA_{11,\overline{1}}=0$$ and $$iR_{,11}=3RA_{11}+6iA_{11,0}-4A_{11,\overline{1}1}.$$

By cancelling $R_{,11}$, one derives$$3RA_{11}+6iA_{11,0}-3A_{11,\overline{1}1}=0.$$

On the other hand, it follows from the commutation relation ([@l]) that$$A_{11,1\overline{1}}-A_{11,\overline{1}1}=iA_{11,0}+2RA_{11},$$

we obtain$$-3RA_{11}+2A_{11,1\overline{1}}-3A_{11,\overline{1}1}=0$$ and then$$-2{\displaystyle \int_{M}}
\left \vert A_{11,1}\right \vert ^{2}d\mu+3{\displaystyle \int_{M}}
\left \vert A_{11,\overline{1}}\right \vert ^{2}d\mu=3{\displaystyle \int_{M}}
R\left \vert A_{11}\right \vert ^{2}d\mu.$$ Moreover, if $R$ is positive constant, then $$\text{\ }A_{11,\overline{1}}=0$$ and $$-2{\displaystyle \int_{M}}
\left \vert A_{11,1}\right \vert ^{2}d\mu=3{\displaystyle \int}
R\left \vert A_{11}\right \vert ^{2}$$ which implies $$A_{11}=0.$$ It 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 $i$ and $-i$, and the corresponding eigenspaces will be denoted by $T^{1,0}$ and $T^{0,1}$, respectively. The integrability condition can then be reformulated as $$X,Y\in T^{1,0} \text{ implies } [X,Y]\in T^{1,0}.$$

Now consider a closed $2n+1$-manifold $M$ with a cooriented contact structure $\xi=\ker \theta$. This means that $\theta \wedge d\theta^{n} \neq0 $. The **Reeb vector field** of $\theta$ is the vector field $T$ uniquely determined by the equations $${\theta}(T)=1,\quad \text{and}\quad d{\theta}(T,{\cdot})=0.\label{eq:defining_Reeb}$$

A **pseudohermitian manifold** is a triple $(M^{2n+1},\theta,J),$ where $\theta$ is a contact form on $M$ and $J$ is a CR structure on $\ker \theta$. The **Levi form** $\left \langle \ ,\  \right \rangle $ is the Hermitian form on $T^{1,0}$ defined by $$H(Z,W)=\left \langle Z,W\right \rangle =-i\left \langle d\theta,Z\wedge
\overline{W}\right \rangle .$$

We can extend this Hermitian form $\left \langle \ ,\  \right \rangle $ to $T^{0,1}$ by defining $\left \langle \overline{Z},\overline{W}\right \rangle
=\overline{\left \langle Z,W\right \rangle }$ for all $Z,W\in T^{1,0}$. Furthermore, the Levi form naturally induces a Hermitian form on the dual bundle of $T^{1,0}$, and hence on all induced tensor bundles.

We now restrict ourselves to **strictly pseudoconvex** CR manifolds, or in other words compatible complex structures $J$. This means that the Levi form induces a Hermitian metric $\langle \cdot,\cdot \rangle_{J,{\theta}}$ by $$\langle V,U\rangle_{J,{\theta}}=d{\theta}(V,JU).$$ The associated norm is defined as usual: $|V|_{J,\theta}^{2}=\langle
V,V\rangle_{J,{\theta}}$. It follows that $H$ also gives rise to a Hermitian metric for $T^{1,0}$, and hence we obtain Hermitian metrics on all induced tensor bundles. By integrating this Hermitian metric over $M$ with respect to the volume form $d\mu=\theta \wedge d\theta^{n}$, we get an $L^{2}$-inner product on the space of sections of each tensor bundle.

The **pseudohermitian connection** or **Tanaka-Webster connection** ([@ta], [@we]) of $(J,\theta)$ is the connection $\nabla$ on $TM\otimes \mathbb{C}$ (and extended to tensors) given in terms of a local frame $\{Z_{\alpha}\}$ for $T^{1,0}$ by $$\nabla Z_{\alpha}=\omega_{\alpha}{}^{\beta}\otimes Z_{\beta}, \quad \nabla
Z_{\bar{ \alpha}}=\omega_{\bar{\alpha}}{}^{\bar{\beta}}\otimes Z_{\bar{\beta}},\quad \nabla T=0,$$ where $\omega_{\alpha}{}^{\beta}$ is the $1$-form uniquely determined by the following equations: $$\begin{split}
d\theta^{\beta} &  =\theta^{\alpha}\wedge \omega_{\alpha}{}^{\beta}+\theta \wedge \tau^{\beta}\\
\tau_{\alpha}\wedge \theta^{\alpha} &  =0\\
\omega_{\alpha}{}^{\beta}+\omega_{\bar{\beta}}{}^{\bar{\alpha}} &  =0.
\end{split}$$ Here $\tau^{\alpha}$ is called the **pseudohermitian torsion**, which we can also write as $$\tau_{\alpha}=A_{\alpha \beta}\theta^{\beta}.$$ The components $A_{\alpha \beta}$ satisfy $$A_{\alpha \beta}=A_{\beta \alpha}.$$ We often consider the **torsion tensor** given by $$A_{J,\theta}=A^{\alpha}{}_{\bar{\beta}}Z_{\alpha}\otimes \theta^{\bar{\beta}}+A^{\bar{\alpha}}{}_{\beta}Z_{\bar{\alpha}}\otimes \theta^{\beta}.$$

We now consider the curvature of the Tanaka-Webster connection in terms of the coframe $\{ \theta=\theta^{0},\theta^{\alpha},\theta^{\bar{\beta}}\} $. The second structure equation gives $$\begin{split}
\Omega_{\beta}{}^{\alpha}  &  =\overline{\Omega_{\bar{\beta}}{}^{\bar{\alpha}}}=d\omega_{\beta}{}^{\alpha}-\omega_{\beta}{}^{\gamma}\wedge \omega_{\gamma}{}^{\alpha},\\
\Omega_{0}{}^{\alpha}  &  =\Omega_{\alpha}{}^{0}=\Omega_{0}{}^{\bar{\beta}}=\Omega_{\bar{\beta}}{}^{0}=\Omega_{0}{}^{0}=0.
\end{split}$$

In [@we Formulas 1.33 and 1.35], Webster showed that the curvature $\Omega_{\beta}{}^{\alpha}$ can be written as $$\begin{array}
[c]{c}\Omega_{\beta}{}^{\alpha}=R_{\beta}{}^{\alpha}{}_{\rho \bar{\sigma}}\theta^{\rho}\wedge \theta^{\bar{\sigma}}+W_{\beta}{}^{\alpha}{}_{\rho}\theta^{\rho}\wedge \theta-W^{\alpha}{}_{\beta \bar{\rho}}\theta^{\bar{\rho}}\wedge \theta+i\theta_{\beta}\wedge \tau^{\alpha}-i\tau_{\beta}\wedge
\theta^{\alpha},
\end{array}
\label{a}$$ where the coefficients satisfy $$\begin{array}
[c]{c}R_{\beta \bar{\alpha}\rho \bar{\sigma}}=\overline{R_{\alpha \bar{\beta}\sigma
\bar{\rho}}}=R_{\bar{\alpha}\beta \bar{\sigma}\rho}=R_{\rho \bar{\alpha}\beta \bar{\sigma}},\  \ W_{\beta \bar{\alpha}\gamma}=W_{\gamma \bar{\alpha}\beta
}.
\end{array}$$ In addition, by [@l (2.4)] the coefficients $W_{\alpha}{}^{\beta}{}_{\rho}$ are determined by the torsion, $$W_{\alpha}{}^{\beta}{}_{\rho}=A_{\alpha \rho,}{}^{\beta}.$$ Contraction of (\[a\]) yields$$\begin{split}
\Omega_{\alpha}{}^{\alpha}=d\omega_{\alpha}{}^{\alpha} &  =R_{\rho \bar{\sigma
}}\theta^{\rho}\wedge \theta^{\bar{\sigma}}+W_{\alpha}{}^{\alpha}{}_{\rho
}\theta^{\rho}\wedge \theta-W_{\overline{\alpha}}{}^{\overline{\alpha}}{}_{\bar{\rho}}\theta^{\bar{\rho}}\wedge \theta \\
&  =R_{\rho \bar{\sigma}}\theta^{\rho}\wedge \theta^{\bar{\sigma}}+A_{\alpha
\rho}{}^{\alpha}\theta^{\rho}\wedge \theta-A_{\bar{\alpha}\bar{\rho}}{}^{\bar{\alpha}}\theta^{\bar{\rho}}\wedge \theta
\end{split}
\label{b}$$

We will denote components of covariant derivatives by indices preceded by a comma. For instance, we write $A_{\alpha \beta,\gamma}$. Here the indices $\{0,\alpha,\bar{\beta}\}$ indicate derivatives with respect to $\{T,Z_{\alpha
},Z_{\bar{\beta}}\}$. For derivatives of a scalar function, we will often omit the comma. For example, $\varphi_{\alpha}=Z_{\alpha}\varphi,\  \varphi
_{\alpha \bar{\beta}}=Z_{\bar{\beta}}Z_{\alpha}\varphi-\omega_{\alpha}{}^{\gamma}(Z_{\bar{\beta}})Z_{\gamma}\varphi,\  \varphi_{0}=T \varphi$ for a (smooth) function $\varphi$.

In particular, we define followings for $n=1$ For a real function $\varphi$, the subgradient $\nabla_{b}$ is defined by $\nabla_{b}\varphi \in \xi$ and $\left \langle Z,\nabla_{b}\varphi \right \rangle _{L_{\theta}}=d\varphi(Z)$ for all vector fields $Z$ tangent to contact plane. Locally $\nabla_{b}\varphi=\varphi_{\bar{1}}Z_{1}+\varphi_{1}Z_{\bar{1}}$. We can use the connection to define the subhessian as the complex linear map $$(\nabla^{H})^{2}\varphi:T_{1,0}\oplus T_{0,1}\rightarrow T_{1,0}\oplus
T_{0,1},$$ by $$(\nabla^{H})^{2}\varphi(Z)=\nabla_{Z}\nabla_{b}\varphi.\  $$ Also $$\Delta_{b}\varphi=Tr\left(  (\nabla^{H})^{2}\varphi \right)  =(\varphi
_{1\bar{1}}+\varphi_{\bar{1}1}).$$ For all $Z=x^{1}Z_{1}\in T_{1,0}$, we define $$\begin{split}
Ric(Z,Z) &  =Wx^{1}x^{\bar{1}}=W|Z|_{L_{\theta}}^{2},\\
Tor(Z,Z) &  =2Re\ iA_{\bar{1}\bar{1}}x^{\bar{1}}x^{\bar{1}}.
\end{split}$$

We also need the following commutation relations ([@l]).$$\begin{array}
[c]{ccl}C_{I,01}-C_{I,10} & = & C_{I,\overline{1}}A_{11}-kC_{I,}A_{11,\overline{1}},\\
C_{I,0\overline{1}}-C_{I,\overline{1}0} & = & C_{I,1}A_{\overline{1}\overline{1}}-kC_{I,}A_{\overline{1}\overline{1},1},\\
C_{I,1\overline{1}}-C_{I,\overline{1}1} & = & iC_{I,0}+kWC_{I}.
\end{array}
\label{6}$$ Here $C_{I}$ denotes a coefficient of a tensor with multi-index $I$ consisting of only $1$ and $\bar{1}$, and $k$ is the number of $1$’s minus the number of $\bar{1}$’s in $I$.

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