Bounded imbeddings of open Kähler manifolds in \({\mathbb{C}}^ n\)
type: Item
instance of: scholarly article
title: Bounded imbeddings of open Kähler manifolds in \({\mathbb{C}}^ n\)
zbMATH Open document ID: 0669.32010
DOI: 10.1215/S0012-7094-89-05810-9
author: John S. Bland
published in: Duke Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: The paper is devoted to the problem of inner characterization of the class of submanifolds of strongly pseudoconvex domains in \({\mathbb{C}}^ N\). It is known that if M is the interior of a compact complex submanifold in \({\mathbb{C}}^ N\) with a smooth strongly pseudoconvex boundary and \(\Phi\) is its smooth bounded strongly pseudoconvex defining function, then the complete Kähler metric with potential (-log-\(\Phi)\) has asymptotically constant negative holomorphic sectional curvature [\textit{P. F. Klembeck}, Indiana Univ. Math. J. 27, 275-282 (1978; Zbl 0422.53032)]. In the previous paper [Invent. Math. 81, 555-566 (1985; Zbl 0588.32001)] the author has shown between otherwise that in fact there is much more stronger analogy with the asymptotic behaviour of the geometry of hyperbolic ball. It is given here the definition of a complete Kähler manifold with asymptotically hyperbolic geometry of order k \((k>1)\). It is shown (Theorem 1), that such a manifold is compactifiable with a \(C^{k-1}\) strongly pseudoconvex boundary. Another characterization has pure function theoretic nature. Author introduces the notion of smoothness (of order k) and strongly pseudoconvexity at infinity of bounded exhausting function on a complex manifold. Theorem 2 says that the manifold with such a function is compactifiable with a (k-2)-smooth strongly pseudoconvex boundary (see also closely connected previous results of \textit{S. Nakano} and \textit{T. Oshawa}, Publ. Res. Inst. Math. Sci. 20, 705-715 (1984; Zbl 0582.32025). Combining these theorems with some known facts, author gives the desired characterizations mentioned above (theorem 4).
Mathematics Subject Classification ID: 32T99
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 32J05
Mathematics Subject Classification ID: 32C25
Mathematics Subject Classification ID: 32Q45
zbMATH DE Number: 4095870
zbMATH Keywords: strongly pseudoconvex submanifold
zbMATH Keywords: Kähler manifold
zbMATH Keywords: asymptotically hyperbolic
zbMATH Keywords: exhausting function
reviewed by: Mikhail Zaidenberg
MaRDI profile type: Publication
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full work available at URL: s0012-7094-89-05810-9
OpenAlex ID: W106778333
description: scientific article; zbMATH DE number 4095870