The K\"ahler-Ricci soliton on bounded pseudoconvex domains

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