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

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 \(S^2 \times S^2\).