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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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References:
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