Rigidity of complete K\"ahler-Einstein metrics under cscK perturbations

In this paper, we study constant scalar curvature K\"ahler (cscK) metrics on complete non-compact K\"ahler--Einstein manifolds. We give sufficient conditions under which a cscK perturbation of a K\"ahler--Einstein metric must remain K\"ahler--Einstein. As a model case, we prove that the Bergman metric on a bounded strictly pseudoconvex domain is K\"ahler--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.