Some applications of Bell's theorem to weakly pseudoconvex domains
The author develops some consequences of Bell's theorem, which states that if \(\Omega \subset \mathbb{C}^ n\) is a bounded, pseudoconvex domain of finite type, if \(\sigma_ n\), \(\sigma \in \Aut (\Omega)\) and if \(\sigma_ n \to \sigma\) on compact sets in \(\Omega\), then \(\sigma_ n \to \sigma\) in \(C^ \infty (\overline \Omega)\). In particular, the author generalizes a theorem of Krantz by showing that for a bounded, pseudoconvex domain satisfying condition \(R\), for \(p\) a boundary point, and for \(\sigma\) an elliptic automorphism, \(\sigma (z) = z + 0 (| z - p |)\) if and only if \(\sigma\) is the identity map. The author also obtains a result on the fixed points of automorphisms of weakly pseudoconvex domains: if \(\Omega\) is pseudoconvex and satisfies condition \(R\) and if \(\overline \Omega\) is diffeomorphic to the closed ball, then every elliptic automorphism of \(\Omega\) has an interior fixed point. Also, a result is obtained concerning the minimal property of the rank of the Levi form at a boundary accumulation point \(p\) of the automorphism group of a bounded, pseudoconvex domain \(\Omega\) of finite type. The author shows that either the rank of the Levi form at \(p\) is minimal over \(\partial \Omega\) or there exists only one point in \(\partial \Omega\) for which the rank of the Levi form is less than it is at \(p\). It is conjectured that in fact only the first situation can occur. The paper concludes with a discussion of some directions for further research.
scientific article; zbMATH DE number 509106
scientific article