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

Abstract

The Chekanov torus was 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 × S 2 and a monotone Lagrangian torus that had been introduced before Chekanov's construction [Che96]. We prove that the monotone Lagrangian torus fiber in a certain Gelfand-Zeitlin system is Hamiltonian isotopic to the Chekanov torus in S 2 × S 2 .