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Parabolic exhaustions for strictly convex domains. (English) Zbl 0581.32018
Given \(D\subset {\mathbb{C}}^ n\) a strictly convex domain and \(p\in D\) any point, the author defines a quasi-strictly parabolic exhaustion \(\tau\) of D with center p. The construction uses essentially the work of L. Lempert on the Kobayashi metric [see Bull. Soc. Math. Fr. 109, 427-474 (1981; Zbl 0492.32025)].
Then to D and p are associated in a canonical way a complete circular domain \(G\subset {\mathbb{C}}^ n\) and a homeomorphism \(h: \bar G\to \bar D\) such that \(h(0)=p\), called the circular representation. The main result is: The circular representation is biholomorphic iff the Monge-Ampère foliation induced by the quasi-strictly parabolic exhaustion \(\tau\) of D centered at p is holomorphic.
As a consequence a characterisation of the strictly convex domains D biholomorphic to a circular domain or to a ball in terms of the properties of the automorphism group of D is obtained.
Reviewer: N.Mihalache

MSC:
32E05 Holomorphically convex complex spaces, reduction theory
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32U05 Plurisubharmonic functions and generalizations
32M05 Complex Lie groups, group actions on complex spaces
32F45 Invariant metrics and pseudodistances in several complex variables
32E10 Stein spaces
Citations:
Zbl 0492.32025
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References:
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