A complete complex hypersurface in the ball of $\mathbb{C}^N$

Abstract

In 1977, P. Yang asked whether there exist complete immersed complex submanifolds $\varphi \colon M^k\rightarrow \mathbb{C}^N$ 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 $\mathbb{B}_N$ of $\mathbb{C}^N$ whose real part is unbounded on every path in $\mathbb{B}_N$ of finite length that ends on $b\mathbb{B}_N$. A consequence is the existence of a complete, closed complex hypersurface in $\mathbb{B}_N$. 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.