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On holomorphic maps between domains in \({\mathbb{C}}^ n\). (English) Zbl 0611.32022
On utilise un résultat récent de T. J. Barth [Proc. Am. Math. Soc. 88, 527-530 (1983; Zbl 0494.32008)] pour montrer qu’une application analytique d’un domaine pseudoconvexe (resp.: convexe dont l’adhérence n’a pas de point extrémal complexe) équilibré, dans un autre domaine ayant les mêmes propriétés, est un isomorphisme linéaire si elle laisse fixe l’origine et conserve la métrique infinitésimale de Kobayashi à l’origine (resp.: la pseudodistance de Kobayashi entre l’origine et un point quelconque du \(1^{er}\) domaine). Passant aux domaines fortement convexes (i.e. de la forme \(\rho <0\) avec \(\rho\in {\mathcal C}^{\infty}\), de \(Hessienne>0\) en tout point où \(\rho =0)\), on obtient des résultats analogues à l’aide de la théorie de L. Lempert [Bull. Soc. Math. France 109, 427-474 (1981; Zbl 0492.32025)].
Reviewer: M.Hervé

MSC:
32H99 Holomorphic mappings and correspondences
32F45 Invariant metrics and pseudodistances in several complex variables
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32T99 Pseudoconvex domains
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