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Extremal plurisubharmonic functions and invariant pseudodistances. (English) Zbl 0584.32037
For any connected open set \(\Omega \subset {\mathbb{C}}^ n\) the author introduces a plurisubharmonic function \(u_{\Omega}\) which can be regarded as a counterpart of the generalized Green’s function with pole at a given point. It is proved that the function \(u_{\Omega}\) decreases under holomorphic mappings of domains in reference, which generalizes the classical Lindelöf property of Green’s functions of one complex variable. Estimates for \(u_{\Omega}\) are obtained in terms of the Carathéodory and Kobayashi pseudodistances on \(\Omega\), and it is proved that \(u_{\Omega}\) satisfies the generalized Monge-Ampère equation. Finally an invariant pseudodistance on \(\Omega\) is defined using the function \(u_{\Omega}\), and some basic properties of the pseudodistance are studied.
Reviewer: K.Shibata

MSC:
32U05 Plurisubharmonic functions and generalizations
32F45 Invariant metrics and pseudodistances in several complex variables
31C10 Pluriharmonic and plurisubharmonic functions
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
32H99 Holomorphic mappings and correspondences
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