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A characterization of hyperbolic manifolds. (English) Zbl 0773.32018
Let \(\text{Hol}(\Delta,X)\) denote the set of holomorphic maps from the open unit disk \(\Delta\) into the (connected, second countable) complex manifold \(X\), and let \(X^*\) denote the one-point compactification of the underlying locally compact Hausdorff space. The author proves that \(X\) is Kobayashi hyperbolic if and only if \(\text{Hol}(\Delta,X)\) is a relatively compact subset of the space of continuous maps from \(\Delta\) into \(X^*\), endowed with the compact-open topology. This characterization resembles (and is equivalent to) the concept “schwach hyperbolisch” defined by W. Kaup [Ann. Inst. Fourier 18(1968), No. 2, 303-330 (1969; Zbl 0174.130)].

MSC:
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
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