The Kähler-Ricci soliton on bounded pseudoconvex domains

In this paper, we study Kähler-Ricci solitons on bounded pseudoconvex domains in ${\mathbb{C}}^n$ with $C^2$ boundary. Under suitable assumptions, we prove that such solitons must be Kähler-Einstein. Building on Huang and Xiao’s resolution of Cheng’s conjecture, we further establish an analogous result for Bergman Kähler-Ricci solitons. Several model domains are presented to illustrate our results.