Lagrangian Spheres in the Complex Euclidean Space Satisfying a Geometric Equality

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For a Lagrangian submanifold of ℂn with scalar curvature τ and mean curvature vector H, the inequality

holds, and the equality is given only in open sets of the Lagrangian subspaces of ℂn or of the Whitney sphere. In this paper, a one-parameter family of Lagrangian spheres including the Whitney sphere is constructed. They satisfy a geometric equality of type τ = η | H | 2, with η >0, and they are globally characterized inside the family of compact Lagrangian submanifolds with null first Betti number in ℂn

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