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

In this paper we construct a properly embedded holomorphic disc in the unit ball B 2 of C 2 having a surprising combination of properties: on the one hand, it has finite area and hence is the zero set of a bounded holomorphic function on B 2 ; on the other hand, its boundary curve is everywhere dense in the sphere b B 2 . A similar result is proved in higher dimensions. Our construction is based on an approximation result in contact geometry, also proved in the paper.