A complete complex hypersurface in the ball of C^N

Abstract: In 1977 P.Yang asked whether there exist complete immersed complex submanifolds g : M^k --> C^N with bounded image. A positive answer is known for holomorphic curves (k=1) and partial answers are known for the case when k>1. The principal result of the present paper is a construction of a holomorphic function on the open unit ball B_N of C^N whose real part is unbounded on every path in B_N of finite length that ends on the boundary of B_N. A consequence is the existence of a complete, closed, complex hypersurface in B_N. This gives a positive answer to Yang's question in all dimensions k, N, 1\leq k<N, by providing properly embedded complete complex manifolds.