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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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References:

[1] T.J. Barth , The Kobayashi indicatrix at the center of a circular domain , Proc. Amer. Math. Soc. , 88 ( 1983 ), pp. 527 - 530 . MR 699426 | Zbl 0494.32008 · Zbl 0494.32008 · doi:10.2307/2045007
[2] R. Braun - W. Kaup - H. Upmeier , On the automorphisms of circular and Reinhardt domains in complex Banach spaces , Manuseripta Math. , 25 ( 1978 ), pp. 97 - 133 . Article | MR 500878 | Zbl 0398.32001 · Zbl 0398.32001 · doi:10.1007/BF01168604
[3] T. Franzoni - E. Vesentini , Holomorphic maps and invariant distances , Math. Studies , 40 , North Holland , Amsterdam , 1980 . MR 563329 | Zbl 0447.46040 · Zbl 0447.46040
[4] I. Graham - H. Wu , Characterization of the unit ball Bn in complex Euclidean Space , Math. Z. , 189 ( 1985 ), pp. 449 - 456 . Article | MR 786275 | Zbl 0547.32013 · Zbl 0547.32013 · doi:10.1007/BF01168151
[5] L. Lempert , La métrique de Kobayashi et la représentation des domaines sur la boule , Bull. Soc. Math. France , 109 ( 1981 ), pp. 427 - 474 . Numdam | MR 660145 | Zbl 0492.32025 · Zbl 0492.32025
[6] L. Lempert , Holomorphic retracts and intrinsic metrics in convex domains , Anal. Math. , 8 ( 1982 ), pp. 257 - 261 . MR 690838 | Zbl 0509.32015 · Zbl 0509.32015 · doi:10.1007/BF02201775
[7] G. Patrizio , Parabolic exhaustions for strictly convex domains , Manuscripta Math. , 47 ( 1984 ), pp. 271 - 309 . Article | MR 744324 | Zbl 0581.32018 · Zbl 0581.32018 · doi:10.1007/BF01174598
[8] G. Patrizio - P.M. Wong , Stability of the Monge-Ampère foliation , Math. Ann. , 263 ( 1983 ), pp. 13 - 29 . Article | MR 697327 | Zbl 0512.32013 · Zbl 0512.32013 · doi:10.1007/BF01457080
[9] W. Rudin , Function theory in the unit ball in Cn , Grund. Math. , 253 , Springer Verlag , New York , 1980 . MR 601594 | Zbl 0495.32001 · Zbl 0495.32001
[10] H. Royden - P.M. Wong , Carathéodory and Kobayashi metric in convex domains , preprint.
[11] A. Sadullaev , Schwarz lemma for circular domains and its applications , Math. Notes , 27 ( 1980 ), pp. 120 - 125 . MR 568402 | Zbl 0448.32011 · Zbl 0448.32011 · doi:10.1007/BF01143011
[12] W. Stoll , A characterization of strictly parabolic manifolds , Ann. Scuola Norm. Sup. Pisa Cl. Sci. , 7 ( 1980 ), pp. 81 - 154 . Numdam | MR 577327 | Zbl 0438.32005 · Zbl 0438.32005
[13] C. Stanton , A characterization of the ball by its intrinsic metrics , Math. Ann. , 264 ( 1983 ), pp. 271 - 275 . Article | MR 711883 | Zbl 0501.32002 · Zbl 0501.32002 · doi:10.1007/BF01457530
[14] E. Vesentini , Complex geodesics , Compositio Math. , 44 ( 1981 ), pp. 375 - 394 . Numdam | MR 662466 | Zbl 0488.30015 · Zbl 0488.30015
[15] J.P. Vigué , Le group des automorphismes analitiques d’un domaine borné d’un espace de Banach complexe. Application aux domaines bornés symmetriques , Ann. Sci. École Norm. Sup. , 9 ( 1976 ), pp. 203 - 282 . Numdam | MR 430335 · Zbl 0333.32027
[16] J.P. Vigué , Caractérisation des automorphismes analytiques d’un domaine convexe borné , C. R. Acad. Sc. Paris , 299 ( 1984 ), pp. 101 - 104 . MR 756530 | Zbl 0589.32042 · Zbl 0589.32042
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