Rigidity of complete Kähler-Einstein metrics under cscK perturbations

Abstract

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.

19 pages. Major revision, including the statement of Theorem 1.2(2)