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arXiv · 2510.13278

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

Zehao Sha
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Abstract

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.

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Zehao Sha. 2025-10-15. Rigidity of complete K\"ahler-Einstein metrics under cscK perturbations. https://arxiv.org/abs/2510.13278

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