Rigidity of complete Kähler-Einstein metrics under cscK perturbations

In this paper, we study constant scalar curvature Kähler (cscK) metrics on complete non-compact Kähler–Einstein manifolds. We give sufficient conditions under which a cscK perturbation of a Kähler–Einstein metric must remain Kähler–Einstein. As a model case, we prove that the Bergman metric on a bounded strictly pseudoconvex domain is Kähler–Einstein whenever it has constant scalar curvature. In particular, combined with Huang–Xiao’s resolution of Cheng’s conjecture, this yields the ball characterization for smooth bounded strictly pseudoconvex domains.