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.