In one complex variable, a bounded domain in the complex plane carries two natural invariant Hermitian metrics: the Bergman metric and the canonical metric (a complete Hermitian metric with constant Gaussian curvature). When the domain has sufficiently smooth boundary so that the Bergman metric is complete, a consequence of a classical result of Qi-Keng Lu shows that these two metrics coincide if and only if the domain is the unit disk.

In higher dimensions, the Bergman metric can be defined analogously for bounded domains. However, the existence of a complete Kähler metric with constant holomorphic sectional curvature is generally too much to expect. Instead, Cheng–Yau and Mok–Yau proved the existence of a complete Kähler metric with constant Ricci curvature (the Kähler–Einstein metric) on such domains when they are pseudoconvex. The Bergman metric reflects the function theory and holomorphic geometry of the domain, while the Kähler–Einstein metric captures its pluripotential and complex geometric structure.
In this talk, I will discuss recent joint work with S. Y. Li (UC Irvine), M. Xiao (UCSD), and Hsiao (Taiwan)–Li (Wuhan), as well as related work by Ebenfelt–Treuner–Xiao and Savale–Xiao. We address longstanding questions concerning when the Bergman metric has constant holomorphic sectional curvature and when it is Einstein.