Rigidity of complete Kähler-Einstein metrics under cscK perturbations

Abstract: In this paper, we study constant scalar curvature K\"ahler (cscK) metrics on a complete non-compact K\"ahler-Einstein manifold. We give a sufficient condition under which any cscK perturbation of the K\"ahler-Einstein metric remains K\"ahler-Einstein. As a model case, we extend Huang-Xiao's resolution of Cheng's conjecture if the Bergman metric has constant scalar curvature on bounded strictly pseudoconvex domains with smooth boundary.