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

Ruh-Vilms Theorems For Minimal Surfaces Without Complex Points and Minimal Lagrangian Surfaces in $\mathbb C P^2$