Proper holomorphic self-mappings of the minimal ball

Nabil Ourimi¹

¹ ICTP Math. Section Strada Costiera, II 34014 Trieste, Italy and Faculté des sciences de Bizerte 7021 Jarzouna, Tunisie

Annales Polonici Mathematici, Tome 79 (2002) no. 2, pp. 97-107

Résumé

The purpose of this paper is to prove that proper holomorphic self-mappings of the minimal ball are biholomorphic. The proof uses the scaling technique applied at a singular point and relies on the fact that a proper holomorphic mapping $f: D\rightarrow {\mit \Omega }$ with branch locus $V_f$ is factored by automorphisms if and only if $f_{*}(\pi _1(D\setminus f^{-1} (f(V_f)), x))$ is a normal subgroup of $\pi _1({\mit \Omega } \setminus f(V_f), b)$ for some $b\in {\mit \Omega } \setminus f(V_f)$ and $x\in f^{-1}(b)$.

Détail

DOI : 10.4064/ap79-2-1

Keywords: purpose paper prove proper holomorphic self mappings minimal ball biholomorphic proof uses scaling technique applied singular point relies proper holomorphic mapping rightarrow mit omega branch locus factored automorphisms only * setminus normal subgroup mit omega setminus mit omega setminus