Complete minimal Kähler surfaces in \(\mathbb{R}^6\) (Q1381942)

Let \(M^n\) be a complete Kähler manifold of complex dimension \(n\geq 2\) and \(f:M^n\longrightarrow {\mathbb R}^{2n+2}\) a minimal isometric immersion of \(M^n\) in flat Euclidean space with real codimension 2. It is known that if we assume \(f\) is not holomorphic in \({\mathbb R}^{2n+2}\equiv {\mathbb C}^{n+1}\), then \(f\) must be globally complex ruled for \(n\geq 3\). The main purpose of this paper is to consider some immersions for \(n=2\). The authors construct examples to show that there exists a class of complete minimal real Kähler surfaces \(M^2\) in \({\mathbb R}^6\) which are irreducible and neither holomorphic nor complex ruled.