On discs in bidiscs (Q2310501)

Let \(\mathbb D\) denote the unit disc in \(\mathbb C\) and let \(\mathbb B\) denote the unit ball in \(\mathbb C^2\). The author proves that there is no continuous map \(F\colon \bar{\mathbb D} \to \bar{\mathbb D}\times \bar{\mathbb D}\), holomorphic on \(\mathbb D\), such that \(F(b\mathbb D)=b(\mathbb D\times \mathbb D)\). The result is in contrast to the previous result by the same author: there is a continuous map \(F\colon \bar{\mathbb D} \to \bar{\mathbb B}\), holomorphic on \(\mathbb D\), such that \(F(b\mathbb D)=b\mathbb B\) [Math. Z. 198, No. 1, 143--150 (1988; Zbl 0652.32015)].