Interpolation by proper holomorphic embeddings of the disc into \(\mathbb{C}^2\)
Let \(\Delta\) be the open unit disc in \(\mathbb C\). A map \(f: \Delta\mathbb\to \mathbb C^2\) is called a proper holomorphic embedding if it is a holomorphic immersion which is one to one and such that the preimage of every compact set is compact. It is known that given a discrete set \(E\subset\mathbb C^2\) there is a proper holomorphic embedding \(f:\Delta\to\mathbb C^2\) such that \(E\subset f(\Delta)\). The author proves a stronger result: Given a discrete set \(S\subset\Delta\) and a proper injection \(\varphi: S\to\mathbb C^2\) there is a proper holomorphic embedding \(f:\Delta\to\mathbb C^2\) that extends \(\varphi\). In other words, given an injective sequence \(\{\zeta_j\}\subset\Delta\) such that \(|\zeta_j|\to 1\) and an injective sequence \(\{w_j\}\subset\mathbb C^2\) such that \(|w_j|\to +\infty\) there is a proper holomorphic embedding \(f:\Delta\to\mathbb C^2\) such that \(f(\zeta_j) = w_j\), \(j\in\mathbb N\). The proof of the Carleman approximation theorem of Buzzard and Forstnerič can be adapted to prove such a result for proper holomorphic embeddings \(f:\mathbb C\to\mathbb C^2\). In the proof one uses the fact that \(\mathbb C\) admits particularly simple embeddings into \(\mathbb C^2\) of the form \(\zeta\to (\zeta,a(\zeta))\) where \(a\) is an entire function. There are no such embeddings for \(\Delta\) so a different proof is necessary in the considered case. In the induction step of the proof, the author uses simultaneous composition by automorphisms on the left and on the right, a novelty introduced by Buzzard and Forstnerič.
simultaneous composition by automorphisms
scientific article; zbMATH DE number 1886051