Rigidity of Extremal and cscK Bergman Metrics on Pseudoconvex Domains

Abstract

Let $\Omega\subset\mathbb C^n$, $n\ge2$, be a bounded connected pseudoconvex domain whose boundary contains a smooth strongly pseudoconvex point. We prove that if the Bergman metric of $\Omega$ is extremal, then its scalar curvature is identically $-n$, and that constant scalar curvature forces the Bergman metric to be K\"ahler--Einstein. Consequently, extremality, constant scalar curvature, and the K\"ahler--Einstein condition are equivalent in this setting. A key analytic ingredient is a local unique-continuation theorem for the Bergman Laplacian at an ACH boundary, proved using H\"ormander's Carleman estimate. We also obtain an extension for possibly unbounded pseudoconvex domains: if the Bergman metric is cscK or extremal near such a boundary point, then it is well-defined and K\"ahler--Einstein throughout the locus where $K_\Omega(z,z)>0$.