COMPLETE DENSELY EMBEDDED COMPLEX LINES IN C-2

In this paper we construct a complete injective holomorphic immersion C -> C-2 whose image is dense in C-2. The analogous result is obtained for any closed complex submanifold X subset of C-n for n > 1 in place of C subset of C-2. We also show that if X intersects the unit ball B-n of C-n and K is a connected compact subset of X boolean AND B-n, then there is a Runge domain Omega subset of X containing K which admits a complete injective holomorphic immersion Omega -> B-n whose image is dense in B-n.