Yau’s conjecture for complete Bergman–Einstein metrics

Determine whether a bounded pseudoconvex domain carrying a complete Bergman–Einstein metric must be homogeneous, as proposed in Yau’s conjecture.

Background

The paper places its Reinhardt-domain rigidity theorem in the broader context of Cheng’s and Yau’s conjectures concerning the geometric consequences of the Kähler–Einstein condition for Bergman metrics. Yau’s formulation concerns bounded pseudoconvex domains and predicts that the existence of a complete Bergman–Einstein metric forces homogeneity. The present paper establishes the corresponding rigidity result for smooth Reinhardt domains, but the quoted formulation concerns the broader conjectural setting beyond that class.