Rigidity of Extremal and cscK Bergman Metrics on Pseudoconvex Domains

Let $Ω\subset\mathbb C^n$, $n\ge2$, be a bounded connected pseudoconvex domain whose boundary contains a smooth strongly pseudoconvex point. We prove that if the Bergman metric of $Ω$ is extremal, then its scalar curvature is identically $-n$, and that constant scalar curvature forces the Bergman me