1209.4737 (Jake P. Solomon)

The Calabi homomorphism, Lagrangian paths and special Lagrangians    [PDF]

Jake P. Solomon

We define a functional $\CC$ on orbits $\OO$ of the group of Hamiltonian symplectomorphisms acting on the space of Lagrangian submanifolds of a symplectic manifold $(X,\omega).$ The functional $\CC$ depends on a differential form $\beta$ of middle degree satisfying $\beta \wedge \omega = 0$ and an exactness condition. In the absence of the exactness condition, $\CC$ is well defined on the universal cover $\widetilde\OO.$ Choosing $\beta$ appropriately, $\CC$ 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 a subspace $\OO^+ \subset \OO.$ As a prerequisite, we study the geodesics of $\OO^+.$

View original: http://arxiv.org/abs/1209.4737