In this article, we investigate Reeb dynamics on $b^m$-contact manifolds, previously introduced in [MiO], which are contact away from a hypersurface $Z$ but satisfy certain transversality conditions on $Z$. The study of these contact structures is motivated by that of contact manifolds with boundary. The search of periodic Reeb orbits on those manifolds thereby starts with a generalization of the well-known Weinstein conjecture. Contrary to the initial expectations, examples of compact $b^m$-contact manifolds without periodic Reeb orbits outside $Z$ are provided. Furthermore, we prove that in dimension $3$, there are always infinitely many periodic orbits on the critical set if it is compact. We prove that traps for the $b^m$-Reeb flow exist in any dimension. This investigation goes hand-in-hand with the Weinstein conjecture on non-compact manifolds having compact ends of convex type. In particular, we extend Hofer's arguments to open overtwisted contact manifolds that are $\mathbb R^+$-invariant in the open ends, obtaining as a corollary the existence of periodic $b^m$-Reeb orbits away from the critical set. The study of $b^m$-Reeb dynamics is motivated by well-known problems in fluid dynamics and celestial mechanics, where those geometric structures naturally appear. In particular, we prove that the dynamics on positive energy level-sets in the restricted planar circular three-body problem are described by the Reeb vector field of a $b^3$-contact form that admits an infinite number of periodic orbits at the critical set.
In this paper, using pseudo-holomorphic curve method, one proves the Weinstein conjecture in the product $P_1\times P_2$ of two strongly geometrically bounded symplectic manifolds under some conditions with $P_1$. In particular, if $N$ is a closed manifold or a noncompact manifold of finite topological type, our result implies that the Weinstein conjecture in $\mathbb{C}\mathbb{P}^2\times T^*N$ holds.
The Weinstein conjecture in the presence of submanifolds having a Legendrian foliation
Helmut Hessel在1993年引入了一种基于全纯曲线的新技术来证明Weinstein猜想。在这些方法适用的情况中，所有接触3-流形$(M,ξ)$与$π_2(M) \ne 0$。我们修改了Hessen的论点，证明了一些高维接触流形的例子的Weinstein猜想。特别是，我们能够证明与实射影空间的连通和总是具有封闭的可收缩Reeb轨道。
We prove the strong Weinstein conjecture for closed contact manifolds that appear as the concave boundary of a symplectic cobordism admitting an essential local foliation by holomorphic spheres.
We propose a theory of contact invariants and open string invariants, assuming that the almost complex $J$ is either non-degenerate or of Bott-type. We do not choose the complex structure $\tilde{J}$ such that $L_X\tilde{J}=0$ on periodic orbits.
A foliation is said to admit a foliated contact structure if there is a codimension 1 distribution in the tangent space of the foliation such that the restriction to any leaf is contact. We prove a version of the Weinstein conjecture in the presence of an overtwisted leaf. The result is shown to be sharp.
In this paper, we introduce the notions of an iterated planar Lefschetz fibration and an iterated planar open book decomposition and prove the Weinstein conjecture for contact manifolds supporting an open book that has iterated planar pages. For $n\geq 1$, we show that a $(2n+1)$-dimensional contact manifold $M$ supporting an iterated planar open book decomposition satisfies the Weinstein conjecture.
How does one measure the failure of Hochschild homology to commute with colimits? Here I relate this question to a major open problem about dynamics in contact manifolds -- the assertion that Reeb orbits exist and are detected by symplectic homology. More precisely, I show that for polarizably Weinstein fillable contact manifolds, said property is equivalent to the failure of Hochschild homology to commute with certain colimits of representation categories of tree quivers. So as to be intelligible to algebraists, I try to include or black-box as much of the geometric background as possible.
Motivated by Poincaré's orbits going to infinity in the (restricted) three-body (see [26] and [6]), we investigate the generic existence of heteroclinic-like orbits in a neighbourhood of the critical set of a $b$-contact form. This is done by using the singular counterpart [3] of Etnyre--Ghrist's contact/Beltrami correspondence [9], and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck [29]. Specifically, we analyze the $b$-Beltrami vector fields on $b$-manifolds of dimension $3$ and prove that for a generic asymptotically exact $b$-metric they exhibit esca…
We prove that if a compact Kähler Poisson manifold has a symplectic leaf with finite fundamental group, then after passing to a finite étale cover, it decomposes as the product of the universal cover of the leaf and some other Poisson manifold. As a step in the proof, we establish a special case of Beauville's conjecture on the structure of compact Kähler manifolds with split tangent bundle.
The Chekanov-Eliashberg dg-algebra is an algebraic invariant of Legendrian submanifolds of contact manifolds, whose definition recently has been extended to singular Legendrians. We describe a way of constructing simpler models of this dg-algebra for singular Legendrian knots in $\mathbb{R}^{3}$, and give the first examples of singular knots for which the full cohomology can be computed. This includes the $A_{n}$-Legendrians. An important question in the study of the Chekanov-Eliashberg dg-algebra is to understand how its quasi-isomorphism type changes when the Legendrian undergoes Weinstein r…
We prove that Lagrangian cocores and Lagrangian linking disks of a stopped Weinstein manifold generate the Lagrangian cobordism infinity-category. As a geometric consequence, we see that any brane (after stabilization) admits a Lagrangian cobordism to a disjoint union of some standard collection of branes (cocores, linking disks, and a zero object). For example, when our stopped Weinstein manifold is a point stopped by itself, we find that any exact brane in Euclidean space admits a Lagrangian cobordism to a disjoint union of cotangent fibers and a zero object. (This is a stronger statement th…
We prove that every symplectic 4-manifold admits a trisection that is compatible with the symplectic structure in the sense that the symplectic form induces a Weinstein structure on each of the three sectors of the trisection. Along the way, we show that a (potentially singular) symplectic braided surface in $\mathbb{CP}^2$ can be symplectically isotoped into bridge position.
我们证明了切触同调区分了任意可定向的、环面的、不可约的3-流形上的无穷多个紧切触结构。作为接触同调计算的结果，在一个非常大的类的环面流形，所有已知的例子普遍紧切触结构与非零挠满足温斯坦猜想。- 关于montre que l 'homologie de contact distingue une infinite de structures de contact tendues sur toute variete toroidale irreductible et orientable de dimensions trois。通过对一个三大类变截面环的接触的同调计算，证明了温斯坦的猜想与接触结构的普遍扭转趋势无关。
Motivated by recent developments in proving the Weinstein conjecture we introduce the notion of covering contact connected sum for virtually contact manifolds and construct virtually contact structures on boundaries of subcritical handle bodies.
We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs, and apply this invariant to various problems in algebraic and symplectic geometry. On the algebro-geometric side, we introduce a new method based on symplectic flexibility and mirror symmetry to bound the Rouquier dimension of derived categories of coherent sheaves on certain complex algebraic varieties and stacks. These bounds are sharp in dimension at most $3$. As a result, we resolve a well-known conjecture of Orlov for a large class of new examples, including all toric $3$-folds and certain log C…
We prove that on closed Riemannian manifolds with infinite abelian, but not cyclic, fundamental group, any isometry that is homotopic to the identity possesses infinitely many invariant geodesics. We conjecture that the result remains true if the fundamental group is infinite cyclic. We also formulate a generalization of the isometry-invariant geodesics problem, and a generalization of the celebrated Weinstein conjecture: on a closed contact manifold with a selected contact form, any strict contactomorphism that is contact-isotopic to the identity possesses an invariant Reeb orbit.
The Chekanov-Eliashberg dg-algebra is a holomorphic curve invariant associated to Legendrian submanifolds of a contact manifold. We extend the definition to Legendrian embeddings of skeleta of Weinstein manifolds. Via Legendrian surgery, the new definition gives direct proofs of wrapped Floer cohomology push-out diagrams. It also leads to a proof of a conjectured isomorphism between partially wrapped Floer cohomology and Chekanov-Eliashberg dg-algebras with coefficients in chains on the based loop space.
We prove that for a broad class of exact symplectic manifolds including ${\mathbb R}^{2m}$ the Hamiltonian flow on a regular compact energy level of an autonomous Hamiltonian cannot be uniquely ergodic. This is a consequence of the Weinstein conjecture and an observation that a Hamiltonian structure with non-vanishing self-linking number must have contact type. We apply these results to show that certain types of exact twisted geodesic flows cannot be uniquely ergodic.
Let $M$ be a symplectic manifold and $G$ a connected, compact Lie group acting on $M$ in a Hamiltonian way. In this paper, we study the equivariant cohomology of $M$ represented by basic differential forms, and relate it to the cohomology of the Marsden-Weinstein reduced space via certain residue formulae using resolution of singularities and the stationary phase principle. In case that $ M $ is a compact, symplectic manifold or the co-tangent bundle of a $G$-manifold, similar residue formulae were derived by Jeffrey, Kirwan et al. for general equivariantly closed forms and by Ramacher for bas…
