In 1977 P. Yang asked whether there exist complete immersed complex sub- manifolds $\varphi: M^k → \mathbb{C}^N$ with bounded image. A positive answer is known for holomorphic curves $(k = 1)$ and partial answers are known for the case when $k > 1$. In the talk we will describe how to construct a holomorphic function on the open unit ball $\mathbb{B}_N$ of $\mathbb{C}^N$ , $N \geq 2$, whose real part is unbounded on every path in $\mathbb{B}_N$ of finite length that ends on $b\mathbb{B}_N$. This implies the existence of a complete, closed complex hypersurface in $\mathbb{B}_N$, and gives a positive answer to Yang’s question in all dimensions $k$, $N$, $1 \leq k < N$, by providing properly embedded complete complex manifolds.