The Calabi homomorphism, Lagrangian paths, and special Lagrangians

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  Calabi defined a homomorphism from the group of Hamiltonian symplectomorphisms to the real numbers.Calabis homomorphism generalizes to a functional on the space of Lagrangian submanifolds Hamiltonian isotopic to a fixed one.In Calabi-Yau manifolds,the functionals critical points are special Lagrangians,and it is convex with respect to a natural metric.The curvature of the metric is non-positive.These results are motivated by mirror symmetry.
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