Turkish Journal of Mathematics




Let N^n(4c) be the complex space form of constant holomorphic sectional curvature 4c, \varphi: M \to 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 \to 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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