The Calabi homomorphism, Lagrangian paths and special Lagrangians

Abstract

Let $\OO$ be an orbit of the group of Hamiltonian symplectomorphisms acting on the space of Lagrangian submanifolds of a symplectic manifold $(X,\omega ).$ We define a functional $\CC:\OO \to \R$ for each differential form $\beta$ of middle degree satisfying $\beta \wedge \omega =0$ and an exactness condition. If the exactness condition does not hold, $\CC$ is defined on the universal cover of $\OO.$ A particular instance of $\CC$ recovers the Calabi homomorphism. If $\beta$ is the imaginary part of a holomorphic volume form, the critical points of $\CC$ are special Lagrangian submanifolds. We present evidence that $\CC$ 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 $\CC$ is convex on an open subspace $\OO^+ \subset \OO.$ As a prerequisite, we define a Riemannian metric on $\OO^+$ 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.

36 pages, fixed minor errors, expanded introduction, added references

References in corpus (1)

• Singularities of Lagrangian mean curvature flow: zero-Maslov class case

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