Convergence and multiplicities for the Lempert function

Given a domain $\Omega \subset \mathbb C$, the Lempert function is a functional on the space Hol (\D,\Omega) of analytic disks with values in $\Omega$, depending on a set of poles in $\Omega$. We generalize its definition to the case where poles have multiplicities given by local indicators (in the sense of Rashkovskii's work) to obtain a function which still dominates the corresponding Green function, behaves relatively well under limits, and is monotonic with respect to the indicators. In particular, this is an improvement over the previous generalization used by the same authors to find an example of a set of poles in the bidisk so that the (usual) Green and Lempert functions differ.Comment: 24 pages; many typos corrected thanks to the referee of Arkiv for Matemati

Pluricomplex Green and Lempert functions for equally weighted poles

For $\Omega$ a domain in $\mathbb C^n$, the pluricomplex Green function with poles $a_1, ...,a_N \in \Omega$ is defined as $G(z):=\sup \{u(z): u\in PSH_-(\Omega), u(x)\le \log \|x-a_j\|+C_j \text{when} x \to a_j, j=1,...,N \}$. 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 $\Omega$ by $L_S (z) :=\inf \{\sum^N_{j=1}\nu_j\log|\zeta_j|: \exists \phi\in \mathcal {O}(\mathbb D,\Omega), \phi(0)=z, \phi(\zeta_j)=a_j, j=1,...,N \}$. It is known that we always have $L_S (z) \ge G_S(z)$. In the more general case where we allow weighted poles, there is a counterexample to equality due to Carlehed and Wiegerinck, with $\Omega$ 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 $L_S (z) > G_S(z)$ 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.Comment: 25 page

The Lempert function and the pluricomplex Green function are not equal in the bidisc

We give a counterexample to Coman's conjecture statingthat in convex domains the pluripolar Green function with several poles and weights equals the corresponding Lempert function

Le cone des fonctions plurisousharmoniques negatives et une conjecture de Coman

Les fonctions plurisousharmoniques n'egatives dans un domaine Omega de C^n forment un c^one convexe. Nous consid'erons les points extr'emaux de ce c^one, et donnons trois exemples. En particulier, nous traitons le cas de la fonction de Green pluricomplexe. Nous calculons celle du bidisque, lorsque les p^oles se situent sur un axe. Nous montrons que cette fonction ne se confonde pas avec la fonction de Lempert correspondante. Cela donne un contre-exemple `a une conjecture de Dan Coman