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.
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I was recently reading this nice survey by AmarzayaOhnita, 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... 

