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

Let $N^n$(4c) be the complex space form of constant holomorphic sectional curvature $4c, varphi : M → N^n(4c)$ be an immersion of an n-dimensional Lagrangian manifold M in $N^n$(4c). Denote by S and H the square of the length of the second fundamental form and the mean curvature of M. Let $rho$ be the non-negative function on M defined by $rho^2=S-nH^2$, 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 $U(varphi) = int _M rho ^2dv = int_M(S − nH^2)dv$ . We call the critical points of $U(varphi)$ the Extremal submanifold in complex space form $N^n$(4c) . We shall get the new Euler-Lagrange equation of $U(varphi)$ and prove some integral inequalities of Simons’ type for n-dimensional compact Extremal Lagrangian submanifolds $varphi : M → N^n(4c)$ in the complex space form $N^n$(4c) in terms of $rho^2, Q,H$ and give some rigidity and characterization Theorems.

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