Symplectic embeddings of polydisks

Hind, Richard; Lisi, Samuel

doi:10.1007/s00029-013-0146-2

Cited by 17 publications

(20 citation statements)

References 22 publications

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“…By the choices made in the splitting construction, this plane is moreover asymptotic to a geodesic on L for the flat metric. The totality of the one-dimensional family of such planes is shown to form a smoothly embedded solid torus having boundary equal to L. Here we need the automatic transversality result [55] by C. Wendl together with the asymptotic intersection results shown in [28] by R. Hind and S. Lisi. We also make heavy use of positivity of intersection for pseudoholomorphic curves; see the work [36] by D. McDuff.…”

Section: 23mentioning

confidence: 99%

“…The splitting construction applied to the unit normal bundle of a Lagrangian submanifold has previously been used in [16,Theorem 1.7.5], [27], [14], [28], and [10], among others. Manifestly, it is an efficient tool for obtaining strong obstructions to Lagrangian embeddings inside uniruled symplectic manifolds.…”

Section: The Splitting Constructionmentioning

confidence: 99%

“…This index computation has been carried out in e.g. [28,Appendix A]. One can either perform a direct computation, or use the fact that the Robbin-Salamon index relates to the Morse index ι µ and nullity ι ν of the corresponding geodesics on L via the formula…”

Section: Index Computations and The Proof Of Theorem Cmentioning

confidence: 99%

“…The following crucial result was proven in [28] by R. Hind and S. Lisi, and uses ideas due to R. Siefring and C. Wendl [46] concerning the asymptotic intersection numbers in the Bott setting. In particular, we refer to [28, Lemma 6.2 & Appendix A].…”

Section: 2mentioning

confidence: 99%

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Lagrangian isotopy of tori in $${S^2\times S^2}$$ S 2 × S 2 and $${{\mathbb{C}}P^2}$$ C P 2

2016

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Abstract. We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic fourmanifolds: the symplectic vector space R 4 , the projective plane CP 2 , and the monotone S 2 × S 2 . The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for T * T 2 , i.e. it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.

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Section: 23mentioning

confidence: 99%

Section: The Splitting Constructionmentioning

confidence: 99%

Section: Index Computations and The Proof Of Theorem Cmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

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Lagrangian isotopy of tori in $${S^2\times S^2}$$ S 2 × S 2 and $${{\mathbb{C}}P^2}$$ C P 2

2016

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“…For example, if there is a symplectic embedding P (1, 1) → E(a, 2a), then ECH capacities only imply that a 1, but the Ekeland-Hofer capacities imply that a 3 2 ; see [5,Remark 1.8]. Another example is that if there is a symplectic embedding from P (1,2) into the ball B(c), then both ECH capacities and Ekeland-Hofer capacities only imply that c 2; but in fact it was recently shown by Hind-Lisi [3] that c 3. In particular, the inclusions P (1, 1) → E( 3 2 , 3) and P (1, 2) → B (3) are 'optimal' in the following sense.…”

mentioning

confidence: 99%

Symplectic embeddings into four-dimensional concave toric domains

Choi

Cristofaro-Gardiner

Frenkel

et al. 2014

Journal of Topology

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ECH (embedded contact homology) capacities give obstructions to symplectically embedding one symplectic four-manifold with boundary into another. We compute the ECH capacities of a large family of symplectic four-manifolds with boundary, called 'concave toric domains'. Examples include the (nondisjoint) union of two ellipsoids in R 4 . We use these calculations to find sharp obstructions to certain symplectic embeddings involving concave toric domains. For example: (1) we calculate the Gromov width of every concave toric domain; (2) we show that many inclusions of an ellipsoid into the union of an ellipsoid and a cylinder are 'optimal'; and (3) we find a sharp obstruction to ball packings into certain unions of an ellipsoid and a cylinder.Contents c k (X, rω) = rc k (X, ω).

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Stabilized Symplectic Embeddings

Hind

2017

Complex and Symplectic Geometry

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Given two four-dimensional symplectic manifolds, together with knots in their boundaries, we define an "anchored symplectic embedding" to be a symplectic embedding, together with a two-dimensional symplectic cobordism between the knots (in the four-dimensional cobordism determined by the embedding). We use techniques from embedded contact homology to determine quantitative critera for when anchored symplectic embeddings exist, for many examples of toric domains. In particular we find examples where ordinarily symplectic embeddings exist, but they cannot be upgraded to anchored symplectic embeddings unless one enlarges the target domain.

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Symplectic embeddings of polydisks

Cited by 17 publications

References 22 publications

Lagrangian isotopy of tori in $${S^2\times S^2}$$ S 2 × S 2 and $${{\mathbb{C}}P^2}$$ C P 2

Lagrangian isotopy of tori in $${S^2\times S^2}$$ S 2 × S 2 and $${{\mathbb{C}}P^2}$$ C P 2

Symplectic embeddings into four-dimensional concave toric domains

Stabilized Symplectic Embeddings

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