For bounded real-analytic pseudoconvex domains the Bergman metric is Kähler-Einstein if and only if the domain is biholomorphic to the unit ball.
Forstnerič, Extending proper holomorphic mappings of positive codimension
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Localization of Bergman Kernels and the Cheng-Yau Conjecture on Real Analytic Pseudoconvex Domains
For bounded real-analytic pseudoconvex domains the Bergman metric is Kähler-Einstein if and only if the domain is biholomorphic to the unit ball.