Kähler-Ricci solitons on bounded pseudoconvex domains with C² boundary are Kähler-Einstein under suitable assumptions, with an analogous result for Bergman versions.
Cao, Existence of gradient K \"a hler- R icci solitons , Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994) 1 (1996), 16
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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.