Let O be an orbit of the group of Hamiltonian symplectomorphisms acting on the space of Lagrangian submanifolds of a symplectic manifold (X,ω) We define a functional C: O→ℝ for each differential form β of middle degree satisfying βΛω=0 and an exactness condition. If the exactness condition does not hold, C is defined on the universal cover of O. A particular instance of C recovers the Calabi homomorphism. If β is the imaginary part of a holomorphic volume form, the critical points of C are special Lagrangian submanifolds. We present evidence that C is related by mirror symmetry to a functional introduced by Donaldson to study Einstein-Hermitian metrics on holomorphic vector bundles. In particular, we show that C is convex on an open subspace O+⊂ O. As a prerequisite, we define a Riemannian metric on O+ and analyze its geodesics. Finally, we discuss a generalization of the flux homomorphism to the space of Lagrangian submanifolds, and a Lagrangian analog of the flux conjecture.
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The author would like to thank G. Tian for introducing him to the subject of geometric stability and for his constant encouragement. Also, the author would like to thank L. Polterovich, E. Shelukhin, R. Thomas and E. Witten for helpful conversations. The author was partially supported by Israel Science Foundation Grant 1321/2009 and Marie Curie Grant No. 239381.