A properly embedded holomorphic disc in the ball with finite area and dense boundary (Q1741835)

The author proves the following (surprising) theorem. For any \(n\geq2\) and \(\varepsilon>0\) there exists a proper holomorphic embedding \(F:\mathbb D\longrightarrow\mathbb B^n\), where \(\mathbb D\) stands for the unit disc and \(\mathbb B^n\) is the unit Euclidean ball in \(\mathbb C^n\), such that: \(F\) extends to an injective holomorphic immersion \(F:\overline{\mathbb D}\setminus\{-1,1\}\longrightarrow\overline{\mathbb B}^n\), \(\operatorname{Area}(F(\mathbb D))<\varepsilon\), and \(F(\partial\mathbb D\setminus\{-1,1\})\) is dense in \(\partial\mathbb B^n\).