Rigidity of complete Kähler-Einstein metrics under cscK perturbations
Zehao Sha
TL;DR
The paper addresses rigidity of complete Kähler–Einstein metrics under cscK perturbations on noncompact manifolds by identifying a set of sufficient conditions under which a complete cscK perturbation of a KE metric remains KE, notably when the manifold is parabolic or has a positive $L^2$-spectrum and the perturbation satisfies controlled $L^2$ growth. It leverages Omori–Yau maximum principle, bounded geometry, and Green’s function estimates to obtain volume-comparison and trace-type bounds, yielding a bi-Lipschitz relation between the perturbed and background metrics under a finite $F$-energy framework. The work also presents model-case results on bounded strictly pseudoconvex domains, showing that if a Cheng–Yau-type KE metric is perturbed to a cscK metric or if a Bergman metric has constant scalar curvature, the resulting metric is KE; in particular, Bergman cscK implies KE and, with boundary smoothness, the domain is biholomorphic to the ball. Collectively, these results connect noncompact rigidity phenomena to classical KE metrics (Cheng–Yau, Bergman) and extend Huang–Xiao’s resolution of Cheng’s conjecture into the broader setting of constant scalar curvature in a noncompact, complete framework.
Abstract
In this paper, we study constant scalar curvature Kähler (cscK) metrics on a complete non-compact Kähler-Einstein manifold. We give a sufficient condition under which any cscK perturbation of the Kähler-Einstein metric remains Kähler-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.