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

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.