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2023•Differential Geometry and its ApplicationsOpen access

Chekanov torus and Gelfand–Zeitlin torus in S2 × S2

Yoosik Kim

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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 × 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 × S 2 by a symplectomorphism.

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What this paper is about

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 × 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 × S 2 by a symplectomorphism.

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Available 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 × 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 × S 2 by a symplectomorphism.

Key concepts: Torus, Mathematics, Monotone polygon, Clifford torus, Lagrangian, Complex torus, Hamiltonian (control theory), Pure mathematics

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