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In this note, we mainly focus on the existence of pseudo-Einstein contactforms, an upper bound eigenvalue estimate for the CR Paneitz operator and itsapplications to the uniformization theorem for Sasakian space form in anembeddable closed strictly pseudoconvex CR 3-manifold. Firstly, the existenceof pseudo-Einstein contact form is confirmed if the CR 3-manifold is Sasakian.Secondly, we derive an eigenvalue upper bound estimate for the CR Paneitzoperator and obtain the CR uniformization theorem for a class of CR3-manifolds. At the end, under the positivity assumption of the pseudohermitiancurvature, we derive the existence theorem for pseudo-Einstein contact formsand uniformization theorems in a closed strictly pseudoconvex CR 3-manifold ofnonnegative CR Paneitz operator with kernel consisting of the CR-pluriharmonicfunctions and the CR Q-curvature is CR-pluriharmonic.

Pseudo-Einstein structure, eigenvalue estimate for the CR Paneitz operator and its applications to uniformization theorem