Let $\Phi$ 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 $\omega =\frac{ i}{ 2}\partial\bar\partial\Phi$. 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 $h(x)=e^{ -\Phi (z)}$, $x=|z|^2$, 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 $f(x)=1-x$.
Categories: math.DG math.CV