Published in Proceedings of Gökova Geometry-Topology Conference 2015
Title Uniqueness of extremal Lagrangian tori in the four-dimensional disc
Author Georgios Dimitroglou Rizell
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 ∂D4 = S3. It also follows that all such tori are Hamiltonian isotopic to the product torus S11/√2×S11/√2⊂S3. This provides an answer to a question by L. Lazzarini in the four-dimensional case.
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