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In this paper we construct a properly embedded holomorphic disc in the unitball $\mathbb{B}^2$ of $\mathbb{C}^2$ having a surprising combination ofproperties: on the one hand, it has finite area and hence is the zero set of abounded holomorphic function on $\mathbb{B}^2$; on the other hand, its boundarycurve is everywhere dense in the sphere $b\mathbb{B}^2$.

A properly embedded holomorphic disc in the ball with finite area and dense boundary curve