Ruh–Vilms theorems for minimal surfaces without complex points and minimal Lagrangian surfaces in $$\mathbb {C}P^2$$

Abstract

In this paper we investigate surfaces in $$\mathbb {C}P^2$$ without complex points and characterize the minimal surfaces without complex points and the minimal Lagrangian surfaces by Ruh–Vilms type theorems. We also discuss the liftability of an immersion from a surface to $$\mathbb {C}P^2$$ into $$S^5$$ in Appendix A.