Balanced metrics on \(\mathbb C^n\)
Let \(g\) be a Kähler metric on \({\mathbb C}^n\) with a Kähler potential \(\Phi\colon{\mathbb C}^n\to{\mathbb R}\) of the form \(\Phi(z_1,\dots,z_n)=g_1(| z_1|^2)+\cdots+g_n(| z_n|^2)\) and assume that all the holomorphic polynomial functions on \({\mathbb C}^n\) belong to the corresponding Hilbert space \({\mathcal H}_\Phi=\{f\in{\mathcal O}({\mathbb C}^n)\mid \int_{{\mathbb C}^n}| f|^2\text{e}^{-\Phi}\text{d}\mu(z)<\infty\}\), where \(\text{d}\mu(z)\) stands for the Lebesgue measure on \({\mathbb C}^n\). In this case any orthonormal basis \(\{f_j\}_{0\leq j<N}\) in \({\mathcal H}_\Phi\) gives rise to a holomorphic map \(z\mapsto[f_0(z),f_1(z),\dots]\) from \({\mathbb C}^n\) into the complex projective space \({\mathbb C}P^N\) (where \(N=\dim{\mathcal H}_\Phi\in\{1,2,\dots,\aleph_0\}\)). The Kähler metric \(g\) is said to be balanced if it is the pull-back of the corresponding Fubini-Study metric by the above map. If this is the case, then the main theorem of the paper under review says that the metric \(g\) is equal to the Euclidean metric on \({\mathbb C}^n\) up to a biholomorphic isometry. The proof of this fact relies on works by \textit{E.~Calabi} [Ann. Math. (2) 58, 1--23 (1953; Zbl 0051.13103)] and \textit{J.~Miles} and \textit{F.~Wiliamson} [J. Lond. Math. Soc., II. Ser. 33, 110--116 (1986; Zbl 0545.30017)].
scientific article; zbMATH DE number 5134191
scientific article