Hyperbolic imbedding and spaces of continuous extensions of holomorphic maps
Let \(X\) be a (not necessarily relatively compact) complex subspace of the complex space \(Y\). Let \(F_ X\) denote the Kobayashi-Royden pseudometric on \(X\), and let \(E\) be a length function on \(Y\) [in the sense of \textit{S. Lang}, `Introduction to complex hyperbolic spaces' (1987; Zbl 0628.32001)]. Following an idea of \textit{H. L. Royden} [Several complex variables. II, Conf. Univ. Maryland 1970, 125-137 (1971; Zbl 0218.32012)], the authors say that a point \(p \in \overline X\) is a hyperbolic point if there are a neighborhood \(U\) of \(p\) in \(Y\) and a constant \(c > 0\) such that \(F_ X \geq cE\) on \(U \cap X\); they observe that \(X\) is hyperbolically imbedded in \(Y\) if and only if every point of \(\overline X\) is a hyperbolic point. The authors then use various spaces of (limits of) holomorphic maps to characterize hyperbolic points and, thereby, hyperbolic embeddedness. These lead to generalizations of results previously known when \(Y\) is nonsingular or \(X = Y\) or \(X \Subset Y\). Included are theorems of \textit{M. Abate} [Proc. Am. Math. Soc. 117, No. 3, 789-793 (1993; Zbl 0773.32018)], \textit{P. Kiernan} [Math. Ann. 204, No. 3, 203-209 (1973; Zbl 0244.32010)], the second author [Ann. Math., II. Ser. 90, No. 1, 9-22 (1969; Zbl 0179.12103)], and, most notably, \textit{J. Noguchi} [Publ. Res. Inst. Math. Sci. 21, No. 1, 27-46 (1985; Zbl 0583.32061)], where the authors' new proof avoids measure theory.
scientific article; zbMATH DE number 653343
scientific article