Bounded imbeddings of open Kähler manifolds in \({\mathbb{C}}^ n\)
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).
scientific article; zbMATH DE number 4095870