A complete complex hypersurface in the ball of C^N

Abstract

In 1977, P. Yang asked whether there exist complete immersed complex submanifolds ’: 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 BN of C N whose real part is unbounded on every path in BN of nite length that ends on bBN . A consequence is the existence of a complete, closed complex hypersurface in BN . This gives a positive answer to Yang’s question in all dimensions k; N; 1 k < N, by providing properly embedded complete complex manifolds.