2002/06/20 by Pascal J. Thomas, Nguyen Van Trao · 1 citation
Mathematics · #math.CV #msc:32U35
paper · pdf · doi:10.1007/bf02390822
25 pages
arxiv created 2002/06/20 · arxiv updated 2009/11/30
For Ω a domain in \mathbb Cn, the pluricomplex Green function with poles a1, ...,aN ∈ Ω is defined as G(z):=sup \u(z): u∈ PSH-(Ω), u(x)≤ log ‖x-aj‖+Cj when x → aj, j=1,...,N \. When there is only one pole, or two poles in the unit ball, it turns out to be equal to the Lempert function defined from analytic disks into Ω by LS (z) :=inf \∑Nj=1νjlog|ζj|: ∃ ϕ∈ \mathcal O(\mathbb D,Ω), ϕ(0)=z, ϕ(ζj)=aj, j=1,...,N \. It is known that we always have LS (z) ≥ GS(z). In the more general case where we allow weighted poles, there is a counterexample to equality due to Carlehed and Wiegerinck, with Ω equal to the bidisk. Here we exhibit a counterexample using only four distinct equally weighted poles in the bidisk. In order to do so, we first define a more general notion of Lempert function "with multiplicities", analogous to the generalized Green functions of Lelong and Rashkovskii, then we show how in some examples this can be realized as a limit of regular Lempert functions when the poles tend to each other. Finally, from an example where LS (z) > GS(z) in the case of multiple poles, we deduce that distinct (but close enough) equally weighted poles will provide an example of the same inequality. Open questions are pointed out about the limits of Green and Lempert functions when poles tend to each other.