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Mathematische Annalen

Unknottedness of real Lagrangian tori in S 2 × S 2 Joontae Kim 1 Received: 7 April 2020 / Revised: 13 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract We prove the Hamiltonian unknottedness of real Lagrangian tori in the monotone S 2 × S 2 , namely any real Lagrangian torus in S 2 × S 2 is Hamiltonian isotopic to the Clifford torus. The proof is based on a neck-stretching argument, Gromov’s foliation theorem, and the Cieliebak–Schwingenheuer criterion. Mathematics Subject Classification 53D12 · 53D35 · 54H25

1 Introduction An even dimensional smooth manifold M equipped with a closed non-degenerate 2-form ω is a symplectic manifold. By Darboux’s theorem [34, Theorem 3.2.2], symplectic manifolds are locally standard, and hence only global properties in symplectic topology are interesting; in particular, the study of middle dimensional submanifolds along which the symplectic form vanishes, namely Lagrangian submanifolds. In 1986, as one of the first steps in symplectic topology [1, Section 6], Arnold proposed the Lagrangian knot problem asking whether two given Lagrangians are isotopic. As formulated in a survey of Eliashberg–Polterovich [22], there are different flavors of isotopy, namely smooth, Lagrangian and Hamiltonian. Hamiltonian isotopies are Lagrangian, and Lagrangian isotopies are smooth. Two Lagrangians are said to be unknotted if they are isotopic to each other in one of these three ways. A remarkable result of Gromov [25] says that there are no closed exact Lagrangians n d xi ∧ dyi ), and hence the extensive study of Lagrangian tori in R2n in (R2n , i=1 has been made for a long time. Chekanov [9] first constructed a monotone Lagrangian torus in R2n for n ≥ 2, which is Lagrangian isotopic while not Hamiltonian isotopic to the Clifford torus TnClif = ×n S 1 , i.e., products of circles in R2 of equal radius. This

Communicated by Jean-Yves Welschinger.

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Joontae Kim [email protected] School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea

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exhibits that a Lagrangian isotopy cannot be in general deformed into a Hamiltonian isotopy. His result is even more interesting since the classical invariants cannot detect this phenomenon. Indeed, the Audin conjecture [2, Section 6.4], which is proved by Polterovich [37] and Viterbo [40] in dimension 4 and Cieliebak–Mohnke [11] in any dimension, says that the minimal Maslov number (one of the classical invariants) of any Lagrangian torus in R2n is always two, so exotic monotone Lagrangian tori are hard to discover. Auroux [3] constructed infinitely many monotone Lagrangian tori in R6 up to Hamiltonian isotopy, while all of them are Lagrangian isotopic. We refer to the work of Dimitroglou Rizell–Evans [17, Corollary C] about the smooth unknottedness of monotone Lagrangian tori inside R2n for n ≥ 5 odd. Since Lagrangian 2-planes in R4 that are asymptotically linear are trivial by Eliashberg–Polterovich [21] (see also [20]), one may expect