Characterization of Whitney spheres among Lagrangian submanifolds with conformal Maslov form

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In this paper, we give new characterizations of Whitney spheres among n-dimensional compact and non-minimal Lagrangian submanifolds immersed in the complex space form Nn(4c) with constant holomorphic sectional curvature 4c. We first show that an n-dimensional compact and non-minimal Lagrangian submanifolds satisfying a point-wise pinching condition in Nn(4c) is a Whitney sphere. Then we prove that there is a positive constant C(n) depending only on n such that, for a compact and non-minimal submanifold Mn immersed in Nn(4c) with c≥0, if the modified second fundamental form B satisfies ∫M|B|ndμ<C(n), then B≡0 and Mn is a Whitney sphere.