Radial Balanced metrics on the unit disk

TL;DR

This paper characterizes the hyperbolic metric on the unit disk as the unique radial balanced metric of height 3 under certain conditions involving a plurisubharmonic function and an entire extension of a related exponential function.

Contribution

It proves a uniqueness result for the hyperbolic metric among radial balanced metrics of height 3 with specific analytic extension properties.

Findings

The hyperbolic metric is uniquely characterized by the given conditions.

A specific functional equation involving $f(x)=1-x$ is key to the proof.

The result links balanced metrics, plurisubharmonic functions, and entire functions.

Abstract

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 $f(x)=1-x$.