Lagrangian submanifolds in complex space forms attaining equality in a basic inequality

Lagrangian submanifolds appear naturally in the context of classical mechanics. Moreover, they play some important roles in supersymmetric field theories as well as in string theory. Recently, it was proved in Chen and Dillen (2011) [11] that for any Lagrangian submanifold M of a complex space form M ˜ n ( 4 c ) , n ⩾ 3 , of constant holomorphic sectional curvature 4c we have δ ( n − 1 ) ⩽ n − 1 4 ( n H 2 + 4 c ) , where H 2 is the squared mean curvature and δ ( n − 1 ) is a δ-invariant of M (cf. Chen, 2000, 2011 [7] , [10] ). In this paper, we completely classify non-minimal Lagrangian submanifolds of complex space forms M ˜ n ( 4 c ) , c = 0 , 1 , − 1 , which satisfy the equality case of the inequality identically.