Bounded holomorphic embeddings of the unit disk into Banach spaces

It is shown that, in contrast to ?ⁿ, infinite dimensional complex Banach spaces E can possess bounded complex closed submanifolds of positive dimension. If E contains c₀ or L¹/H ₀ ¹ then the unit disk D can be embedded into E as a bounded complex closed submanifold. If, however, E has the analytic Radon-Nikodym property then no bounded embedding exists. Acknowledgement: I thank W. Hensgen and M. Schottenloher for many stimulating discussions.