Lagrangian product tori in symplectic manifolds

Chekanov, Yuri; Schlenk, Félix

doi:10.4171/cmh/391

Cited by 5 publications

(18 citation statements)

References 19 publications

(27 reference statements)

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“…Versal Deformations. Versal deformations were introduced in [8] and subsequently used in [9] and [10] as a tool to distinguish Lagrangian submanifolds. The idea is to look at the behaviour of known symplectic invariants on neighbouring Lagrangians of the submanifolds in question.…”

Section: Versal Deformations Of Real Lagrangiansmentioning

confidence: 99%

“…Since we will only use the displacement energy as an invariant, we will restrict ourselves to this case. We refer to [10] for details.…”

Section: Versal Deformations Of Real Lagrangiansmentioning

confidence: 99%

“…which has a natural Hamiltonian action on C n . Now take a smooth embedded curve γ(t) = r(t)e 2πiϑ(t) in C which encloses area 1 and for which 0 < ϑ(t) < 1 n and 0 < r(t) < n π + δ, (10) for a small δ > 0. From γ construct the curve Γ(t) = 1…”

Section: Application Ii: Chekanov Torimentioning

confidence: 99%

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Real Lagrangian Tori and Versal Deformations

Brendel

2020

Preprint

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Can a given Lagrangian submanifold be realized as the fixed point set of an anti-symplectic involution? If so, it is called real. We give an obstruction for a closed Lagrangian submanifold to be real in terms of the displacement energy of nearby Lagrangians. Applying this obstruction to toric fibres, we obtain that the central fibre of many (and probably all) toric monotone symplectic manifolds is real only if the corresponding moment polytope is centrally symmetric. Furthermore, we embed the Chekanov torus in all toric monotone symplectic manifolds and show that it is exotic and not real, extending Kim's result [16] for S 2 × S 2 . Inside products of S 2 , we show that all products of Chekanov tori are pairwise distinct and not real either. These results indicate that real tori are rare.

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Section: Versal Deformations Of Real Lagrangiansmentioning

confidence: 99%

“…Since we will only use the displacement energy as an invariant, we will restrict ourselves to this case. We refer to [10] for details.…”

Section: Versal Deformations Of Real Lagrangiansmentioning

confidence: 99%

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Real Lagrangian Tori and Versal Deformations

Brendel

2020

Preprint

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“…We briefly outline the adjustments required the establish the obstruction part of Theorem 3. Note that there does not exist an embedding L(1, x) → P (a, b) when a < 1 since by [3], Proposition 2.1, the Lagrangian torus L(1, x) has displacement energy 1. We still argue by contradiction, assuming that a < 2 and b < x.…”

Section: Proof Of the Obstruction Part Of Theoremmentioning

confidence: 99%

Squeezing Lagrangian tori in dimension 4

Hind¹,

Opshtein²

2019

Preprint

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“…product tori) is called exotic. The construction and study of exotic tori is a very active field of research, see for example Chekanov-Schlenk [CS10,CS16], Galkin-Usnich [GA10], Auroux [Aur15], Vianna [Via16,Via17]. In terms of full classification of such Lagrangian submanifolds, very little is known.…”

Section: Lagrangian Submanifoldsmentioning

confidence: 99%

On the topology of Lagrangian submanifolds in toric symplectic manifolds

Brendel¹

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This thesis consists of four main chapters. Chapter 2, An introduction to the Chekanov torus, is an expository article introducing all the concepts necessary to define and under- stand the Chekanov torus. We give a detailed introduction to Hamiltonian circle actions, symplectic reduction, lifting techniques and versal deformations and show how those can be used to show that the Chekanov torus is exotic. Chapter 3, On the topology of real Lagrangians in toric symplectic manifolds, is a joint article with Joontae Kim and Jiyeon Moon focusing on constructing examples of real Lagrangian submanifolds in toric manifolds by lifting symmetries from the moment polytope. We also prove convexity and tightness for the examples we construct and give an analogue of the Delzant construction. Chapter 4, Real Lagrangian tori and versal deformations, is an article focusing on obstructions for a given Lagrangian submanifold to be real and is, in some sense, complementary to Chapter 3. We develop a general obstruction in terms of versal deformations and displacement energy and apply this to toric fibres and Chekanov tori in toric manifolds. Chapter 5, Squeezing via degenerations of the complex projective plane, is an appendix to the paper On certain quantifications of Gromov’s non-squeezing theorem by Kevin Sackel, Antoine Song, Umut Varolgunes and Jonathan J. Zhu. We prove that the symplectic four-ball can be squeezed after removing a subset of Minkowski dimension two.

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Lagrangian product tori in symplectic manifolds

Cited by 5 publications

References 19 publications

Real Lagrangian Tori and Versal Deformations

Real Lagrangian Tori and Versal Deformations

Squeezing Lagrangian tori in dimension 4

On the topology of Lagrangian submanifolds in toric symplectic manifolds

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