The invariant Szegő metric and characterization of the complex ball

On a bounded domain in ℂ 𝑛 with 𝐶 2 -smooth boundary, given a suitable measure on the boundary, the Szegő kernel and the Szegő metric can be defined. When the measure is of some special form, the Szegő metric is an invariant Kähler metric, which generalizes the classical Fefferman-Szegő metric on the bounded smooth strongly pseudoconvex domain. Following the arguments by Fu-Wong, Nemirovski-Shafikov and Huang-Xiao on the Bergman-Einstein metric on bounded smooth strongly pseudoconvex domains, we prove an analogous result for the Fefferman-Szegő-Einstein metric. More precisely, the Fefferman-Szegő metric on a bounded smooth strongly pseudoconvex domain in ℂ 𝑛 is Kähler-Einstein if and only if the domain is biholomorphic to the complex unit ball. Moreover, we further characterize the complex unit ball among bounded smooth strongly pseudoconvex domains in ℂ 𝑛 where the Fefferman-Szegő metric is proportional to one of the Bergman, Carathéodory and Kobayashi metrics.