The following interesting quantity was introduced by K. Cieliebak and K. Mohnke for a Lagrangian submanifold L of a symplectic manifold: the minimal positive symplectic area of a disc with boundary on L. They also showed that this quantity is bounded from above by π/n for a Lagrangian torus inside the 2n-dimensional unit disc equipped with the standard symplectic form. A Lagrangian torus for which this upper bound is attained is called extremal. We show that all extremal Lagrangian tori inside the four-dimensional unit disc are contained in the boundary ∂D^{4} = S^{3}. It also follows that all such tori are Hamiltonian isotopic to the product torus S^{1}_{1/√2}×S^{1}_{1/√2}⊂S^{3}. This provides an answer to a question by L. Lazzarini in the four-dimensional case. |