Extremal Lagrangian submanifolds in a complex space form N^n(4c)

Let Nn(4c) be the complex space form of constant holomorphic sectional curvature 4c , φ : M → 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 ρ be the non-negative function on M defined by ρ^2 = S − nH^2 , 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(φ) = ∫M ρ^2dv = ∫M(S − nH^2)dv . We call the critical points of U(φ) the Extremal submanifold in complex space form Nn(4c) . We shall get the new Euler-Lagrange equation of U(φ) and prove some integral inequalities of Simons’ type for n-dimensional compact Extremal Lagrangian submanifolds φ : M → Nn(4c) in the complex space form Nn(4c) in terms of ρ^2, Q,H and give some rigidity and characterization Theorems