Extremal Lagrangian tori in toric domains

Abstract

Let $L$ be a closed Lagrangian submanifold of a symplectic manifold $(X,\omega )$. Cieliebak and Mohnke define the symplectic area of $L$ as the minimal positive symplectic area of a smooth $2$-disk in $X$ with boundary on $L$. An extremal Lagrangian torus in $(X,\omega )$ is a Lagrangian torus that maximizes the symplectic area among the Lagrangian tori in $(X,\omega )$. We prove that every extremal Lagrangian torus in the symplectic unit ball $({\bar{B}}^{2n}(1),{\omega }_{\mathrm{std}})$ is contained entirely in the boundary $\partial B^{2n}(1)$. This answers a question attributed to Lazzarini and completely settles a conjecture of Cieliebak and Mohnke in the affirmative. In addition, we prove the conjecture for a class of toric domains in $({\mathbb{C}}^n,{\omega }_{\mathrm{std}})$, which includes all compact strictly convex four-dimensional toric domains. We explain with counterexamples that the general conjecture does not hold for non-convex domains.

94 pages, 12 figures, final version. To appear in the Journal of Symplectic Geometry