On plurisubharmonicity of the solution of the Fefferman equation and its applications to estimate the bottom of the spectrum of Laplace-Beltrami operators

In this paper, we introduce a concept of super-pseudoconvex domain. We prove that the solution of the Feffereman equation on a smoothly bounded strictly pseudoconvex domain $D$ in $\CC^n$ is plurisubharmonic if and only if $D$ is super-pseudoconvex. As an application, we give a lower bound estimate the bottom of the spectrum of Laplace-Beltrami operators when $D$ is super-pseudoconvex by using the result of Li and Wang \cite{LiWang}.