Holomorphic vector fields and proper holomorphic self-maps of Reinhardt domains
A theorem due to \textit{H. Alexander} [ Indiana Univ. Math. J. 26, 137--146 (1977; Zbl 0391.32015)] says that \textit{any proper holomorphic self-mapping of the unit ball in \(\mathbb C^{n+1}\), \(n\geq1\), is an automorphism.} Recall that a domain in \(\mathbb C^{n+1}\) is said to be a \textit{Reinhardt domain} (respectively a \textit{complete} Reinhardt domain) if for each of its points \((z_1,\dots, z_{n+1})\) it contains the torus (respectively, polydisc) \((\zeta_1z_1,\dots,\zeta_{n+1}z_{n+1})\), \(| \zeta_i| =1\) (respectively, \(| \zeta_i| \leq1\)) for any \(i\). The paper under review extends Alexander's theorem to arbitrary bounded complete Reinhardt domains in \(\mathbb C^{n+1}\) and arbitrary bounded Reinhardt domains in \((\mathbb C^*)^{n+1}\); in both cases the boundary is supposed to be \(C^2\). The same result is proved for arbitrary bounded complete Reinhardt domains whose boundary is \(C^{\infty}\)- strictly pseudoconvex somewhere. Open question: Does the statement of Alexander's theorem hold true for arbitrary smoothly bounded domains in \(\mathbb C^{n+1}\)? Earlier, Alexander's theorem was extended to certain pseudoconvex domains with regular boundaries and well behaved sets of weakly pseudoconvex points, see \textit{S. I. Pinchuk} [Sov. Math. Dokl. 19, 804--807 (1978; Zbl 0422.32020); translation from Dokl. Akad. Nauk SSSR 241, 30--33 (1978; Zbl 0422.32020)], \textit{E. Bedford} and \textit{S. Steve} [Math. Ann. 261, 47--49 (1982; Zbl 0499.32016)], \textit{K. Diederich} and \textit{John E. Fornaess} [Math. Ann. 259, 279--286 (1982; Zbl 0486.32013)]. For the proof of the results of the paper, the author studies the Lie algebra of holomorphic vector fields tangent to the boundary (which is a Reinhardt hypersurface). In the case when the latter hypersurface is \(C^2\)- strictly pseudoconvex, it is shown that the previous Lie algebra is finite dimensional and consists of rational vector fields on \((\mathbb C^*)^{n+1}\).
scientific article; zbMATH DE number 2062419