Uniqueness of extremal Lagrangian tori in the four-dimensional disc

TL;DR

This paper proves that extremal Lagrangian tori in the four-dimensional unit disc are uniquely characterized as the standard product torus on the boundary, answering a specific question in symplectic topology.

Contribution

It establishes the uniqueness and boundary containment of extremal Lagrangian tori in four dimensions, confirming their Hamiltonian isotopy to a standard product torus.

Findings

Extremal Lagrangian tori lie on the boundary of the four-dimensional disc.

Such tori are Hamiltonian isotopic to the standard product torus.

The result confirms a conjecture for the four-dimensional case.

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 $\pi/n$ 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 $\partial D^4=S^3$, and is hence Hamiltonian isotopic to the product torus $S^1_{1/\sqrt{2}} \times S^1_{1/\sqrt{2}} \subset S^3$. This provides an answer to a question by L. Lazzarini in the four-dimensional case.