Radial Balanced metrics on the unit disk

Abstract

Let $Φ$ be a strictly plurisubharmonic and radial function on the unit disk ${\cal D}\subset {\complex}$ and let $g$ be the \K metric associated to the \K form $ω=\frac{i}{2}\partial\bar\partialΦ$. We prove that if $g$ is $g_{eucl}$-balanced of height 3 (where $g_{eucl}$ is the standard Euclidean metric on ${\complex}={\real}^2$), and the function $h(x)=e^{-Φ(z)}$, $x=|z|^2$, extends to an entire analytic function on ${\real}$, then $g$ equals the hyperbolic metric. The proof of our result is based on a interesting characterization of the function $f(x)=1-x$.