Kähler-Ricci solitons on bounded pseudoconvex domains with C² boundary are Kähler-Einstein under suitable assumptions, with an analogous result for Bergman versions.
Demailly, Complex analytic and differential geometry, https://www-fourier.ujf-grenoble.fr/ demailly/manuscripts/agbook.pdf
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The K\"ahler-Ricci soliton on bounded pseudoconvex domains
Kähler-Ricci solitons on bounded pseudoconvex domains with C² boundary are Kähler-Einstein under suitable assumptions, with an analogous result for Bergman versions.