Holomorphicity of minimal submanifolds in complex space forms (Q1094689)

Let M be a Kähler manifold and let \({\mathbb{C}}Q_ c\) be a complex space form with holomorphic sectional curvature c. The problem of the present paper is to find conditions which imply that a minimal isometric immersion f of M into \({\mathbb{C}}Q_ c\) is holomorphic. The authors give a complete answer to the problem in the case \(c<0:\) If \(\dim_{{\mathbb{C}}} M>1\), then f is holomorphic or antiholomorphic, if \(\dim_{{\mathbb{C}}} M=1\), then there are examples of minimal immersions which are not holomorphic. It is not enough to restrict the dimension of M if \(c>0\), but if f is a circular immersion a similar theorem holds for positively curved complex space forms.