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  • Volume of small balls and sub-Riemannian curvature in 3D contact manifolds
    J. Symplectic Geom. (IF 0.657) Pub Date : 
    Davide Barilari; Ivan Beschastnyi; Antonio Lerario

    We compute the asymptotic expansion of the volume of small sub-Riemannian balls in a contact $3$‑dimensional manifold, and we express the first meaningful geometric coefficients in terms of geometric invariants of the sub-Riemannian structure.

    更新日期:2020-07-20
  • Quantization of Hamiltonian coactions via twist
    J. Symplectic Geom. (IF 0.657) Pub Date : 
    Pierre Bieliavsky; Chiara Esposito; Ryszard Nest

    In this paper we introduce a notion of quantum Hamiltonian (co)action of Hopf algebras endowed with Drinfel’d twist structure (resp., 2-cocycles). First, we define a classical Hamiltonian action in the setting of Poisson Lie groups compatible with the 2-cocycle structure and we discuss a concrete example. This allows us to construct, out of the classical momentum map, a quantum momentum map in the

    更新日期:2020-07-20
  • On linking of Lagrangian tori in $\mathbb{R}^4$
    J. Symplectic Geom. (IF 0.657) Pub Date : 
    Laurent Côté

    We prove some results about linking of Lagrangian tori in the symplectic vector space $(\mathbb{R}^4 , \omega)$. We show that certain enumerative counts of holomophic disks give useful information about linking. This enables us to prove, for example, that any two Clifford tori are unlinked in a strong sense. We extend work of Dimitroglou Rizell and Evans on linking of monotone Lagrangian tori to a

    更新日期:2020-07-20
  • Stability conditions and Lagrangian cobordisms
    J. Symplectic Geom. (IF 0.657) Pub Date : 
    Felix Hensel

    In this paper we study the interplay between Lagrangian cobordisms and stability conditions. We show that any stability condition on the derived Fukaya category $\mathcal{DFuk} (M)$ of a symplectic manifold $(M, \omega)$ induces a stability condition on the derived Fukaya category of Lagrangian cobordisms $\mathcal{DFuk} (\mathbb{C} \times M)$. In addition, using stability conditions, we provide general

    更新日期:2020-07-20
  • On the lower bounds of the $L^2$-norm of the Hermitian scalar curvature
    J. Symplectic Geom. (IF 0.657) Pub Date : 
    Julien Keller; Mehdi Lejmi

    On a pre-quantized symplectic manifold, we show that the symplectic Futaki invariant, which is an obstruction to the existence of constant Hermitian scalar curvature almost-Kähler metrics, is actually an asymptotic invariant. This allows us to deduce a lower bound for the $L^2$-norm of the Hermitian scalar curvature as obtained by S. Donaldson [15] in the Kähler case.

    更新日期:2020-07-20
  • Potential functions on Grassmannians of planes and cluster transformations
    J. Symplectic Geom. (IF 0.657) Pub Date : 
    Yuichi Nohara; Kazushi Ueda

    With a triangulation of a planar polygon with $n$ sides, one can associate an integrable system on the Grassmannian of $2$‑planes in an $n$‑space. In this paper, we show that the potential functions of Lagrangian torus fibers of the integrable systems associated with different triangulations glue together by cluster transformations. We also prove that the cluster transformations coincide with the wall-crossing

    更新日期:2020-07-20
  • Persistence-like distance on Tamarkin’s category and symplectic displacement energy
    J. Symplectic Geom. (IF 0.657) Pub Date : 2020-05-01
    Tomohiro Asano; Yuichi Ike

    We introduce a persistence-like pseudo-distance on Tamarkin’s category and prove that the distance between an object and its Hamiltonian deformation is at most the Hofer norm of the Hamiltonian function. Using the distance, we show a quantitative version of Tamarkin’s non-displaceability theorem, which gives a lower bound of the displacement energy of compact subsets of cotangent bundles.

    更新日期:2020-05-01
  • Loose Legendrian and pseudo-Legendrian knots in $3$-manifolds
    J. Symplectic Geom. (IF 0.657) Pub Date : 2020-05-01
    Patricia Cahn; Vladimir Chernov

    We prove a complete classification theorem for loose Legendrian knots in an oriented $3$-manifold, generalizing results of Dymara and Ding-Geiges. Our approach is to classify knots in a $3$-manifold $M$ that are transverse to a nowhere-zero vector field $V$ up to the corresponding isotopy relation. Such knots are called $V$-transverse. A framed isotopy class is simple if any two $V$-transverse knots

    更新日期:2020-05-01
  • Positive topological entropy of positive contactomorphisms
    J. Symplectic Geom. (IF 0.657) Pub Date : 2020-05-01
    Lucas Dahinden

    A positive contactomorphism of a contact manifold $M$ is the end point of a contact isotopy on $M$ that is always positively transverse to the contact structure. Assume that $M$ contains a Legendrian sphere $\Lambda$, and that $(M, \Lambda)$ is fillable by a Liouville domain $(W, \omega)$ with exact Lagrangian $L$. We show that if the exponential growth of the action filtered wrapped Floer homology

    更新日期:2020-05-01
  • H-principle for complex contact structures on Stein manifolds
    J. Symplectic Geom. (IF 0.657) Pub Date : 2020-05-01
    Franc Forstnerič

    In this paper we introduce the notion of a formal complex contact structure on an odd dimensional complex manifold. Our main result is that every formal complex contact structure on a Stein manifold, $X$, is homotopic to a holomorphic contact structure on a Stein domain $\Omega \subset X$ which is diffeotopic to $X$. We also prove a parametric h‑principle in this setting, analogous to Gromov’s h‑principle

    更新日期:2020-05-01
  • Ideal Liouville Domains, a cool gadget
    J. Symplectic Geom. (IF 0.657) Pub Date : 2020-05-01
    Emmanuel Giroux

    Liouvile domains are central objects in symplectic geometry today, but they have unsatisfactory aspects due to the requested choice of Liouville forms and to the non-compactness of their completions. Ideal Liouville domains, in contrast, are compact manifolds with boundary merely equipped with a symplectic form in the interior. Still, their isomorphism classes are in one-to-one correspondence with

    更新日期:2020-05-01
  • Symplectic and Kähler structures on biquotients
    J. Symplectic Geom. (IF 0.657) Pub Date : 2020-05-01
    Oliver Goertsches; Panagiotis Konstantis; Leopold Zoller

    We construct symplectic structures on roughly half of all equal rank biquotients of the form $G //T$, where $G$ is a compact simple Lie group and $T$ a torus, and investigate Hamiltonian Lie group actions on them. For the Eschenburg flag, this action has similar properties as Tolman’s and Woodward’s examples of Hamiltonian non-Kähler actions. In addition to the previously known Kähler structure on

    更新日期:2020-05-01
  • The fundamental groups of contact toric manifolds
    J. Symplectic Geom. (IF 0.657) Pub Date : 2020-05-01
    Hui Li

    Let $M$ be a connected compact contact toric manifold. Most of such manifolds are of Reeb type. We show that if $M$ is of Reeb type, then $\pi_1 (M)$ is finite cyclic, and we describe how to obtain the order of $\pi_1 (M)$ from the moment map image.

    更新日期:2020-05-01
  • Coisotropic Hofer–Zehnder capacities and non-squeezing for relative embeddings
    J. Symplectic Geom. (IF 0.657) Pub Date : 2020-05-01
    Samuel Lisi; Antonio Rieser

    We introduce the notion of a symplectic capacity relative to a coisotropic submanifold of a symplectic manifold, and we construct two examples of such capacities through modifications of the Hofer–Zehnder capacity. As a consequence, we obtain a non-squeezing theorem for symplectic embeddings relative to coisotropic constraints and existence results for leafwise chords on energy surfaces.

    更新日期:2020-05-01
  • Positive loops of loose Legendrian embeddings and applications
    J. Symplectic Geom. (IF 0.657) Pub Date : 2020-05-01
    Guogang Liu

    In this paper, we prove that there exist contractible positive loops of Legendrian embeddings based at any loose Legendrian submanifold. As an application, we define a partial order on $\widetilde{Cont}_0 (M, \xi)$, called strong orderability, and prove that overtwisted contact manifolds are not strongly orderable.

    更新日期:2020-05-01
  • Spinor modules for Hamiltonian loop group spaces
    J. Symplectic Geom. (IF 0.657) Pub Date : 2020-05-01
    Yiannis Loizides; Eckhard Meinrenken; Yanli Song

    Let $LG$ be the loop group of a compact, connected Lie group $G$. We show that the tangent bundle of any proper Hamiltonian $LG$-space $\mathcal{M}$ has a natural completion $\overline{T}\mathcal{M}$ to a strongly symplectic $LG$-equivariant vector bundle. This bundle admits an invariant compatible complex structure within a natural polarization class, defining an $LG$-equivariant spinor bundle $\

    更新日期:2020-05-01
  • A new approach to the symplectic isotopy problem
    J. Symplectic Geom. (IF 0.657) Pub Date : 2020-05-01
    Laura Starkston

    The symplectic isotopy conjecture states that every smooth symplectic surface in $\mathbb{C}\mathrm{P}^2$ is symplectically isotopic to a complex algebraic curve. Progress began with Gromov’s pseudoholomorphic curves [Gro85], and progressed further culminating in Siebert and Tian’s proof of the conjecture up to degree $17$ [ST05], but further progress has stalled. In this article we provide a new direction

    更新日期:2020-05-01
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