A Complete Complex Hypersurface in the Ball of C^N

In the talk we will describe the recent result of the speaker how to construct a holomorphic function on the open unit ball B_N of C^N, N>1, whose real part is unbounded on every path in B_N of finite length that ends on bB_N. Level sets of such functions are examples of complete complex hypersurfaces in the ball and give a complete answer to a question of P. Yang from 1977 about the existence of bounded complete complex manifods. We will also present a generalization of this result to pseudoconvex domains in C^N and a related result obtained jointly with A. Alarcon and F. J. Lopez about a construction of a complete proper holomorphic embedding from the open unit disc in C to B_2.