Uniqueness of extremal Lagrangian tori in the four-dimensional disc

2015/11/30 by Rizell, Georgios Dimitroglou · 1 citation
#53D12 #53D42 #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.1512.00039

Abstract

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 an extremal Lagrangian torus inside the four-dimensional unit disc is contained in the boundary ∂ D⁴=S³, and is hence Hamiltonian isotopic to the product torus S¹_1/√(2) × S¹_1/√(2) ⊂ S³. This provides an answer to a question by L. Lazzarini in the four-dimensional case.

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