On minimal Lagrangian submanifolds in complex space forms with semi-parallel second fundamental form.

In this paper, we study the interesting open problem of classifying the minimal Lagrangian submanifolds of dimension n in complex space forms with semi-parallel second fundamental form. First, we completely solve the problem in cases n = 2 , 3 , 4. Second, supposing further that the scalar curvature is constant for n ≥ 5 , we also give an answer to the problem by applying the classification theorem of [F. Dillen, H. Li, L. Vrancken and X. Wang, Lagrangian submanifolds in complex projective space with parallel second fundamental form, Pacific J. Math. 255 (2012) 79–115]. Finally, for such Lagrangian submanifolds in the above problem with n ≥ 3 , we establish an inequality in terms of the traceless Ricci tensor, the squared norm of the second fundamental form and the scalar curvature. Moreover, this inequality is optimal in the sense that all the submanifolds attaining the equality are completely determined. [ABSTRACT FROM AUTHOR]