Balanced metrics on CnCn

Let gg be a Kähler metric on CnCn and let HΦHΦ be the complex Hilbert space consisting of global holomorphic functions ff on CnCn such that ∫Cne−Φ|f|2dμ(z)<∞, where Φ:Cn→RΦ:Cn→R is a Kähler potential for gg and dμ(z) is the standard Lebesgue measure on CnCn. In this paper we prove that if (1) gg is balanced with respect to the Euclidean metric, (2) Φ(z)=g1(|z1|2)+⋯+gn(|zn|2)Φ(z)=g1(|z1|2)+⋯+gn(|zn|2) and (3) z1j1⋯znjn belong to HΦHΦ, for all non-negative integers j1,…jnj1,…jn, then, up to biholomorphic isometries, gg equals the Euclidean metric. The proof is based on Calabi’s diastasis function and on the characterization of the exponential function due to Miles and Williamson.