Green vs. Lempert functions: a minimal example

TL;DR

This paper provides a minimal example demonstrating that the Lempert function and the pluricomplex Green function, which coincide for two poles, can differ when three poles are involved in the unit ball.

Contribution

It presents the first known example where the Lempert function and the pluricomplex Green function do not coincide for three poles in the unit ball.

Findings

Lempert function and Green function differ for three poles in the unit ball.

The example refutes the general equality for more than two poles.

Highlights limitations of previous coincidence results.

Abstract

The Lempert function for a set of poles in a domain of $\mathbb C^n$ at a point $z$ is obtained by taking a certain infimum over all analytic disks going through the poles and the point $z$, and majorizes the corresponding multi-pole pluricomplex Green function. Coman proved that both coincide in the case of sets of two poles in the unit ball. We give an example of a set of three poles in the unit ball where this equality fails.