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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
Zbl 0492.32025
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