A remark on spaces of holomorphic maps to taut manifolds (Q1270268)

The Blaschke condition states that for a given discrete subset \(S\) of the unit disk \(\Delta = \{z\in \mathbb C : | z| < 1\}\), there exists a non-constant bounded holomorphic function \(f\in\mathcal H^{\infty}(\Delta)\) with \(f|_S \equiv 0\) iff \(\sum_{z\in S}(1 - | z|) <\infty\). As a consequence, the restriction map \(r: \text{Hol}(\Delta,\Delta)\to \text{Map}(S,\Delta)\) is injective for every discrete subset \(S\in\Delta\) with \(\sum_{z\in S}(1 - | z|) =\infty\). The author proves a similar statement for mappings to taut manifolds. Recall that a complex manifold \(X\) is called taut if \(\text{Hol}(\Delta,X)\) is a normal family. Let \(X, Y\) be taut complex manifolds. Then there exists a discrete subset \(S \subset X\) such that the natural restriction map \(r: \text{Hol}(X, Y)\to \text{Map}(S, Y)\) is injective. Moreover, the image is closed and \(r\) is a homeomorphism onto its image.