Rigidity of Extremal and cscK Bergman Metrics on Pseudoconvex Domains

Let $\mathrm{\Omega }\subset {\mathbb{C}}^n$, $n\ge 2$, be a bounded connected pseudoconvex domain whose boundary contains a smooth strongly pseudoconvex point. We prove that if the Bergman metric of $\mathrm{\Omega }$ is extremal, then its scalar curvature is identically $-n$, and that constant scalar curvature forces the Bergman metric to be Kähler–Einstein. Consequently, extremality, constant scalar curvature, and the Kähler–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 Hörmander’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 Kähler–Einstein throughout the locus where $K_{\mathrm{\Omega }}(z,z)>0$.