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

TL;DR

This paper investigates K"ahler-Ricci solitons on bounded pseudoconvex domains, proving they are K"ahler-Einstein under certain conditions and extending results related to Bergman solitons.

Contribution

It establishes that K"ahler-Ricci solitons on such domains are K"ahler-Einstein and extends prior results to Bergman K"ahler-Ricci solitons.

Findings

K"ahler-Ricci solitons are K"ahler-Einstein under suitable assumptions

Extension of Cheng's conjecture resolution to Bergman solitons

Illustrative model domains provided

Abstract

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.