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Moha Boutat
Quasi-homogénéité des applications holomorphes propres d’un domaine quasi-disqué sur un domaine disqué
Annales de la faculté des sciences de Toulouse Sér. 6, 20 no. 4 (2011), p. 759-765, doi: 10.5802/afst.1322
Article PDF | Reviews MR 2918212 | Zbl 1243.32014

Résumé - Abstract

A result of F. Berteloot and G.Patrizio [1] states that if $f$ is a proper holomorphic map between two bounded complete circular domains $\Omega _1$ and $\Omega _2 $ in ${\bf C^{n+1}} (n\ge 1)$, such that $ f^{-1}\lbrace 0\rbrace =\lbrace 0\rbrace $ and such that the principal part $f_p$ of the Taylor expansions of $f$ at the origin is nondegenerated i.e $ {f_p}^{-1}\lbrace 0\rbrace =\lbrace 0\rbrace $, then $f\equiv f_p$. Here we propose to generalize their result in the case where $\Omega _1$ is a complete quasi-circular domain and $\Omega _2 $ is a complete circular domain. Moreover this proof does not use the tools of projective dynamics of J. E. Fornaess and N. Sibony [3].


[1] Berteloot (F.) and Patrizio (G.).— A Cartan theorem for proper holomorphic mappings of complete circular domains, Advances in math 153, p. 342-352 (2000).  MR 1770933 |  Zbl 0962.32020
[2] Boutat (M.).— Applications holomorphes propres d’un domaine quasi-disqué sur un domaine disqué, Bulletin des Sciences Mathématiques Volume 133, Issue 4, May p. 335-347 (2009).  MR 2532688 |  Zbl 1176.32006
[3] Fornaess (J. E.) and Sibony (N.).— Complex dynamics in higher dimensions, in “Complex Pluripotential theory,”, NATO Adv. Sci. Inst. Ser. C. Math. Phys. Sci., Vol. 439, pp. 131-186, Kluwer Academic, Dordrecht (1994).  MR 1332961 |  Zbl 0811.32019
[4] Barth (T.).— The Kobayashi indicatrix at the center of a circular domain, Proc. Amer. Math. Soc. 88, p. 527-530 (1983).  MR 699426 |  Zbl 0494.32008
[5] Chirka (E.M.), Delbeault (P.), Khenkin (G.M.), Vitushkin (A.G.).— Introduction to complex analysis, Springer-Verlag (1997).  MR 1734390 |  Zbl 0947.32001
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