Balanced metrics on Cn

Let be a Kahler metric on C-n and let H-Phi be the complex Hilbert space consisting of global holomorphic functions f on C" such that integral(Cn)e(-Phi)vertical bar f vertical bar(2)d mu(z) < infinity. where Phi : C-n -> R is a K ihler potential for g and d mu(z) is the standard Lebesgue measure on C-n. In this paper we prove that if (1) g is balanced with respect to the Euclidean metric, (2) Phi(Z) = g(1)(vertical bar z vertical bar (2)) and (3) z(1)(j1)...z(n)(jn) belong to H-Phi, for all non-negative integers j(1).... j(n), then, up to biholomorphic isometrics, g 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. (c) 2006 Elsevier B.V. All rights reserved.