Chekanov torus and Gelfand–Zeitlin torus in S2 × S2

Abstract

The Chekanov torus is the first known 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 that had been constructed before Chekanov's construction [6]. We prove that the monotone Lagrangian torus fiber in a certain Gelfand–Zeitlin system is related to the Chekanov torus in $S^2\times S^2$ by a symplectomorphism.