Chekanov torus and Gelfand--Zeitlin torus in $S^2 \times S^2$

TL;DR

This paper demonstrates that a specific Gelfand--Zeitlin Lagrangian torus in $S^2 imes S^2$ is Hamiltonian isotopic to the Chekanov torus, establishing a deep connection between these two important monotone Lagrangian tori.

Contribution

It proves the Hamiltonian isotopy between the Chekanov torus and a Gelfand--Zeitlin torus in $S^2 imes S^2$, linking two significant constructions in symplectic topology.

Findings

The Gelfand--Zeitlin torus is Hamiltonian isotopic to the Chekanov torus.

This establishes a new relationship between different monotone Lagrangian tori.

The result enhances understanding of the symplectic topology of $S^2 imes S^2$.

Abstract

The Chekanov torus was the first known \emph{exotic} torus, a monotone Lagrangian torus that is not Hamiltonian isotopic to the standard monotone Lagrangian torus. We explore the relationship between the Chekanov torus in $S^2 \times S^2$ and a monotone Lagrangian torus which had been introduced before Chekanov's construction \cite{Chekanov}. We prove that the monotone Lagrangian torus fiber in a certain Gelfand--Zeitlin system is Hamiltonian isotopic to the Chekanov torus in $S^2 \times S^2$.