Holomorphic vector fields and proper holomorphic self-maps of Reinhardt domains

Proper holomorphic self-maps of smooth bounded Reinhardt domains in ℂ2

“…Since Ω is C 2 smooth and bounded, it contains a strictly pseudoconvex point on its boundary, and thereforeΩ contains the same point. Thus a result of Berteloot [1,Theorem 1.3] implies that f is biholomorphic.…”

“…In higher dimensions, we have a weaker result which is a simple consequence of [1] and used in the proof of Theorem 1.4. Since Alexander's theorem, the problem has been solved for many cases.…”

“…In [1], Berteloot proved the following theorem regarding the proper holomorphic self-maps. The novelty here is that neither the pseudoconvexity nor the finite type of the domain is assumed, only the completeness.…”

“…Now we are ready to apply the results of Berteloot [1] using the lemmas above to get the following properties of the self-maps on the branch locus. The following is a special case of Proposition 3.1 of [1], and is the key to control the branch locus.…”

“…We [7] verified the case of the smooth bounded pseudoconvex Reinhardt domains of finite type in C n . Berteloot [1] solved the case of the complete Reinhardt domains with the C 2 smooth boundary (without the pseudoconvexity assumption) in C n . His method is based on the study of the Lie algebra of the holomorphic tangent vector fields of a strictly pseudoconvex Reinhardt hypersurface.…”