Hi, Dear colleague, is there any survey paper on the classification theorems on Lagrangian submanifolds in a Complex space form with parallel mean curvature vector? and any insight from you on this subject is welcome.
I was recently reading this nice survey by Amarzaya-Ohnita, and they have an entire section on the subject you are interested (see Section 3). They claim that totally real submanifolds with parallel mean curvature vector in $\mathbb C^n$ and $\mathbb C P^n$ have been classified by Naitoh and Takeuchi (references 8, 9, 10 & 11 in their bibliography). Maybe you should take a look at those papers too.
As a side note, just googling "Lagrangian submanifold parallel mean curvature vector" you get lots of interesting links...