The Calabi homomorphism, Lagrangian paths and special Lagrangians

TL;DR

This paper introduces a functional on Lagrangian submanifolds related to the Calabi homomorphism, explores its critical points as special Lagrangians, and connects it to mirror symmetry and geometric structures.

Contribution

It defines a new functional on Lagrangian orbits, relates it to the Calabi homomorphism, and investigates its properties and connections to mirror symmetry and geometric analysis.

Findings

The functional generalizes the Calabi homomorphism.

Critical points correspond to special Lagrangian submanifolds.

The functional exhibits convexity on a specific subspace.

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.