Let Nn(4c) be the complex space form of constant holomorphic sectional curvature 4c, j: M \to Nn(4c) be an immersion of an n-dimensional Lagrangian manifold M in Nn(4c). Denote by S and H the square of the length of the second fundamental form and the mean curvature of M. Let r be the non-negative function on M defined by r2=S-nH2, Q be the function which assigns to each point of M the infimum of the Ricci curvature at the point. In this paper, we consider the variational problem for non-negative functional U(j)=\intMr2dv=\intM(S-nH2)dv. We call the critical points of U(j) the Extremal submanifold in complex space form Nn(4c). We shall get the new Euler-Lagrange equation of U(j) and prove some integral inequalities of Simons' type for n-dimensional compact Extremal Lagrangian submanifolds j: M \to Nn(4c) in the complex space form Nn(4c) in terms of r2, Q, H and give some rigidity and characterization Theorems.

Let Nn(4c) be the complex space form of constant holomorphic sectional curvature 4c, j: M \to Nn(4c) be an immersion of an n-dimensional Lagrangian manifold M in Nn(4c). Denote by S and H the square of the length of the second fundamental form and the mean curvature of M. Let r be the non-negative function on M defined by r2=S-nH2, Q be the function which assigns to each point of M the infimum of the Ricci curvature at the point. In this paper, we consider the variational problem for non-negative functional U(j)=\intMr2dv=\intM(S-nH2)dv. We call the critical points of U(j) the Extremal submanifold in complex space form Nn(4c). We shall get the new Euler-Lagrange equation of U(j) and prove some integral inequalities of Simons' type for n-dimensional compact Extremal Lagrangian submanifolds j: M \to Nn(4c) in the complex space form Nn(4c) in terms of r2, Q, H and give some rigidity and characterization Theorems.