On Legendrian foliations in contact manifold I: Singularities and neighborhood theorems

横截结的邻域与不稳定化

Neighborhoods of transverse knots and destabilizations

本文研究横截结的唯一标准邻域及非紧致Legendrian结的结构定理，揭示了Legendrian结的一般不稳定化结果，并发现无Giroux扭转的流形具有无限多个紧致接触结构。

The topology of stable electromagnetic structures and Legendrian fields on the 3-sphere

Null solutions to Maxwell's equations in free space have the property that the topology of the electric and magnetic lines is preserved for all time. In this article we connect the study of a particularly relevant class of null solutions (related to the Hopf fibration) with the existence of pairs of volume preserving Legendrian fields with respect to the standard contact structure on the 3-sphere. Exploiting this connection, we prove that a Legendrian link can be realized as a set of closed orbits of a non-vanishing Legendrian field corresponding to the electric or magnetic part of a null solu…

Engel structures with trivial characteristic foliations

Engel structures on M x S^1 and M x I are studied in this paper, where M is a 3-dimensional manifold. We suppose that these structures have characteristic line fields parallel to the fibres, S^1 or I. It is proved that they are characterized by contact structures on the cross section M, the twisting numbers, and Legendrian foliations on both ends M x dI in the case of M x I.

过扭切触3-流形中带Legendrian边界的凸圆盘

Convex disks with Legendrian boundary in overtwisted contact 3-manifolds

本研究在闭过扭切触3-流形中对带固定特征叶与Legendrian边界的凸圆盘做切触同痕分类，验证其符合h-原理，推导相关Legendrian平凡纽结空间的h-原理性质，并确定过扭3-球面中一类Legendrian纽结补的切触映射类群。

veering triangulations的Legendrian位置

Legendrian position of veering triangulations

本文研究veering triangulations与bicontact结构的关系，证明其边可作为Legendrian弧，为研究伪Anosov流提供工具。

Topological entropy for Reeb vector fields in dimension three via open book decompositions

Given an open book decomposition of a contact three man-ifold (M, $ξ$) with pseudo-Anosov monodromy and fractional Dehn twist coefficient c = k n, we construct a Legendrian knot $Λ$ close to the stable foliation of a page, together with a small Legendrian pushoff $Λ$. When k $\ge$ 5, we apply the techniques of [CH2] to show that the strip Legen-drian contact homology of $Λ$ $\rightarrow$ $Λ$ is well-defined and has an exponential growth property. The work [Al2] then implies that all Reeb vector fields for $ξ$ have positive topological entropy.

Foliation-coupling Dirac structures

We extend the notion of "coupling with a foliation" from Poisson to Dirac structures and get the corresponding generalization of the Vorobiev characterization of coupling Poisson structures. We show that any Dirac structure is coupling with the fibers of a tubular neighborhood of an embedded presymplectic leaf, give new proofs of the results of Dufour and Wade on the transversal Poisson structure, and compute the Vorobiev structure of the total space of a normal bundle of the leaf. Finally, we use the coupling condition along a submanifold, instead of a foliation, in order to discuss submanifo…

高维中的过度扭曲凸超曲面

An overtwisted convex hypersurface in higher dimensions

本文证明了高维中特定凸超曲面周围的接触结构芽是过度扭曲的，并在具有环面核心的 plastikstufe 附近找到此类超曲面，同时附录重新证明了 Legendrian unknot 的松散性及奇异分布决定接触结构芽的民间结论。

Structures de contact sur les varietes fibrees en cercles au-dessus d'une surface

In this paper, we study the global behaviour of contact structures on oriented manifolds V which are circle bundles over a closed orientable surface S of genus g>0. We establish in particular contact analogs of a number of classical results about foliations due to Milnor, Wood, Thurston, Matsumoto, and Ghys. In Section~1, we prove that V carries a (positive) contact structure transverse to the fibers if and only if the Euler number of the fibration is less or equal to 2g-2. In Section~2, we show that, for any contact structure $ξ$ on V, one of the following properties holds: either $ξ$ is isot…

Singularities of integrable Hamiltonian systems: a criterion for non-degeneracy, with an application to the Manakov top

Let (M,ω) be a symplectic 2n-manifold and h_1,...,h_n be functionally independent commuting functions on M. We present a geometric criterion for a singular point P\in M (i.e. such that {dh_i(P)}_{i=1}^n are linearly dependent) to be non-degenerate in the sence of Vey-Eliasson. Then we apply Fomenko's theory to study the neighborhood U of the singular Liouville fiber containing saddle-saddle singularities of the Manakov top. Namely, we describe the singular Liouville foliation on U and the `Bohr-Sommerfeld' lattices on the momentum map image of U. A relation with the quantum Manakov top studied…

与通用闭合二形式相关的分层以及分层 $L_\infty$ 空间

Stratifications associated to generic closed two-forms and stratified $L_\infty$ spaces

本文研究了通用闭合二形式与分层 $L_\infty$ 空间的关系，通过构造分层结构和关联 $L_\infty$ 空间来扩展对 $L_\infty$ 结构的研究。

Non-Exact Symplectic Cobordisms Between Contact 3-Manifolds

We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably less rigid than its counterparts for exact or Stein cobordisms: in particular, we exhibit large new classes of contact 3-manifolds which are symplectically cobordant to something overtwisted, or to the tight 3-sphere, or which admit symplectic caps containing symplectically embedded spheres with vanishing self-intersection. These constructions imply new and simplified proofs of several recent results involving fillability, planarity and non-separating contact type embeddin…