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Non-linearity of the pluricomplex Green function. (English) Zbl 0969.32014
By a deep theorem due to Lempert, the pluricomplex Green function $$g(z,w)$$ of a convex domain $$\Omega\subset\mathbb{C}^n$$ with pole at $$w\in \Omega$$ coincides with the function $$\delta(z,w)= \inf\log|f^{-1}(w)|$$, the infimum being taken over all holomorphic mappings $$f$$ from the unit disc $$\Delta \subset \mathbb{C}$$ to $$\Omega$$ such that $$f(0)=z$$ and $$w\in f(\Delta)$$. To deal with the Green functions with several poles, the author extends the notion of the Lempert function $$\delta$$. Namely, let $$A=\{(w_j, \nu_j)\}_j$$ be a finite subset of $$\Omega\times \mathbb{R}_+$$, then $$\delta(z,A): =\inf\sum_j \nu_j \log |f^{-1} (w_j)|$$, where $$f(0)=z$$ and $$w_j\in f(\Delta)$$. Properties of $$\delta(z,A)$$ and its relations to the Green function $$g(z,A)$$ are studied. In particular, the set $$\{z:\delta (z,A)=\sum_j \nu_j\delta (z,w_j)\}$$ is described as the union of all complex geodesics (for the Kobayashi metric) passing through all the points $$w_j$$.

##### MSC:
 32U35 Plurisubharmonic extremal functions, pluricomplex Green functions 32F45 Invariant metrics and pseudodistances in several complex variables
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##### References:
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