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.
Categories: math.SG math.DG