An example of limit of Lempert Functions

Abstract

The Lempert function for several poles $a_0, ..., a_N$ in a domain $Ω$ of $\mathbb C^n$ is defined at the point $z \in Ω$ as the infimum of $\sum^N_{j=0} \log|ζ_j|$ over all the choices of points $ζ_j$ in the unit disk so that one can find a holomorphic mapping from the disk to the domain $Ω$ sending 0 to $z$. This is always larger than the pluricomplex Green function for the same set of poles, and in general different. Here we look at the asymptotic behavior of the Lempert function for three poles in the bidisk (the origin and one on each axis) as they all tend to the origin. The limit of the Lempert functions (if it exists) exhibits the following behavior: along all complex lines going through the origin, it decreases like $(3/2) \log |z|$, except along three exceptional directions, where it decreases like $2 \log |z|$. The (possible) limit of the corresponding Green functions is not known, and this gives an upper bound for it.