A Note On Weinstein's Conjecture

On the Weinstein conjecture in higher dimensions

The existence of a "Plastikstufe" for a contact structure implies the Weinstein conjecture for all supporting contact forms.

The Weinstein conjecture in the uniruled manifolds

In this note we prove the Weinstein conjecture for a class of symplectic manifolds including the uniruled manifolds based on Liu-Tian's result.

The Weinstein conjecture for iterated planar contact structures

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.

The Weinstein Conjecture in Product of Symplectic Manifolds

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.

Taubes's proof of the Weinstein conjecture in dimension three

This is an introduction to Taubes's proof of the Weinstein conjecture, written for the AMS Current Events Bulletin. It is intended to be accessible to nonspecialists, so much of the article is devoted to background and context.

Some constructions of Weinstein manifolds with chaotic Reeb dynamics

The goal of this note is to describe some constructions of Weinstein manifolds with chaotic Reeb dynamics, and to explain how this property can sometimes be detected directly from the skeleton.

The Weinstein conjecture in the presence of submanifolds having a Legendrian foliation

Helmut Hofer introduced in '93 a novel technique based on holomorphic curves to prove the Weinstein conjecture. Among the cases where these methods apply are all contact 3--manifolds $(M,ξ)$ with $π_2(M) \ne 0$. We modify Hofer's argument to prove the Weinstein conjecture for some examples of higher dimensional contact manifolds. In particular, we are able to show that the connected sum with a real projective space always has a closed contractible Reeb orbit.

The Weinstein conjecture for connected sums

We prove the Weinstein conjecture for non-trivial contact connected sums under either of two topological conditions: non-trivial fundamental group or torsion-free homology.

Strong fillability and the Weinstein conjecture

Extending work of Chen, we prove the Weinstein conjecture in dimension three for strongly fillable contact structures with either non-vanishing first Chern class or with strong and exact filling having non-trivial canonical bundle. This implies the Weinstein conjecture for certain Stein fillable contact structures obtained by the Eliashberg-Gompf construction.For example we prove the Weinstein conjecture for the Brieskorn homology spheres $Σ(2,3,6n-1)$, $n\geq2$, oriented as the boundary of the corresponding Milnor fibre. Furthermore, for tight contact structures on odd lens spaces, non-contra…

Polyfolds, Cobordisms, and the strong Weinstein conjecture

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.

Orderability and the Weinstein Conjecture

In this article we prove that the Weinstein conjecture holds for contact manifolds $(Σ,ξ)$ for which $\mathrm{Cont}_0(Σ,ξ)$ is non-orderable in the sense of Eliashberg-Polterovich [EP00]. More precisely, we establish a link between orderable and hypertight contact manifolds. In addition, we prove for certain contact manifolds a conjecture by Sandon [San13b] on the existence of translated points in the non-degenerate case.

The Weinstein Conjecture for Hamiltonian Fibrations

In this note we extend to non trivial Hamiltonian fibrations over symplectically uniruled manifolds a result of Lu's, \cite{Lu}, stating that any trivial symplectic product of two closed symplectic manifolds with one of them being symplectically uniruled verifies the Weinstein Conjecture for closed separating hypersurfaces of contact type, under certain technical conditions. The proof is based on the product formula for Gromov-Witten invariants ($GW$-invariant) of Hamiltonian fibrations derived in \cite{H}.

切触叶状结构与广义Weinstein猜想

Contact foliations and generalised Weinstein conjectures

该研究将切触流形上的Reeb流推广为高维切触叶状结构，提出两个叶拓扑相关的广义Weinstein猜想，并在和乐保黎曼度量等情形下给出部分正面结果，拓展了切触动力学的已有成果。

The Seiberg-Witten equations and the Weinstein conjecture

Let M denote a compact, oriented 3-manifold and let a denote a contact 1-form on M. This article proves that the vector field that generates the kernel of the 2-form da has at least one closed, integral curve.

Contact Invariants, Open String Invariants and Weinstein Conjecture

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.

The Weinstein conjecture with multiplicities on spherizations

Let M be a smooth closed manifold and T*M its cotangent bundle endowed with the usual symplectic structure. A hypersurface S in T*M is said to be fiberwise starshaped if for each point q in M the intersection of S with the fiber at q is starshaped with respect to the origin. In this thesis we give lower bounds of the growth rate of the number of closed Reeb orbits on a fiberwise starshaped hypersurface in terms of the topology of the free loop space of M. We distinguish the two cases that the fundamental group of the base space M has an exponential growth of conjugacy classes or not. If the ba…

Symplectic cobordisms and the strong Weinstein conjecture

We study holomorphic spheres in certain symplectic cobordisms and derive information about periodic Reeb orbits in the concave end of these cobordisms from the non-compactness of the relevant moduli spaces. We use this to confirm the strong Weinstein conjecture (predicting the existence of null-homologous Reeb links) for various higher-dimensional contact manifolds, including contact type hypersurfaces in subcritical Stein manifolds and in some cotangent bundles. The quantitative character of this result leads to the definition of a symplectic capacity.

The Weinstein conjecture and the theorems of nearby and almost existence

The Weinstein conjecture, as the general existence problem for periodic orbits of Hamiltonian or Reeb flows, has been among the central questions in symplectic topology for over two decades and its investigation has led to understanding of some fundamental properties of Hamiltonian flows. In this paper we survey some recently developed and well-known methods of proving various particular cases of this conjecture and the closely related almost existence theorem. We also examine differentiability and continuity properties of the Hofer-Zehnder capacity function and relate these properties to the …

Open books and the Weinstein conjecture

We show the existence of a contractible periodic Reeb orbit for any contact structure supported by an open book whose binding can be realised as a hypersurface of restricted contact type in a subcritical Stein manifold. A key ingredient in the proof is a higher-dimensional version of Eliashberg's theorem about symplectic cobordisms from a contact manifold to a symplectic fibration.

The Seiberg-Witten equations and the Weinstein conjecture II: More closed integral curves of the Reeb vector field

Let M denote a compact, orientable, 3-dimensional manifold and let a denote a contact 1-form on M; thus the wedge product of a with da is nowhere zero. This article explains how the Seiberg-Witten Floer homology groups as defined for any given Spin-C structure on M give closed, integral curves of the vector field that generates the kernel of da.

An algebraic approach to the algebraic 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.

The Weinstein conjecture for stable Hamiltonian structures

We use the equivalence between embedded contact homology and Seiberg-Witten Floer homology to obtain the following improvements on the Weinstein conjecture. Let Y be a closed oriented connected 3-manifold with a stable Hamiltonian structure, and let R denote the associated Reeb vector field on Y. We prove that if Y is not a T^2-bundle over S^1, then R has a closed orbit. Along the way we prove that if Y is a closed oriented connected 3-manifold with a contact form such that all Reeb orbits are nondegenerate and elliptic, then Y is a lens space. Related arguments show that if Y is a closed orie…

接触瞬子与Weinstein猜想及Arnold弦猜想的证明

Contact instantons and Proofs of Weinstein's conjecture and Arnold's chord conjecture

本文基于接触Hamilton动力学与接触瞬子模空间理论，通过利用Legendrian接触瞬子上同调的基本类存在性、评估横截性以及接触同胚的Legendrian化几何构造，完整证明了Weinstein猜想和Arnold弦猜想。

共形辛Weinstein猜想

A conformal symplectic Weinstein conjecture

本文提出并部分验证了局部共形辛流形上的Weinstein猜想推广，引入Reeb 2-曲线，并利用Gromov-Witten理论构造Fuller指标，获得经典Reeb动力学新结果。

奇异 Weinstein 猜想的一个反例

A counterexample to the singular Weinstein conjecture

本文在 3 维 b-接触流形上构造 Reeb 向量场，证明奇异周期轨道数目可预先指定，并由此给出文献[23]中奇异 Weinstein 猜想与 Hamilton Seifert 猜想的反例，同时指出广义 Weinstein 猜想仍待解决。

Subelliptic Spin_c Dirac operators, III The Atiyah-Weinstein conjecture

In this paper we show that there is a well defined modified dbar-Neumann problem for a spin_c manifold with a strictly pseudoconvex boundary (in the contact geometry sense). We show that the index of the associated boundary value problem can be computed as the relative index between the Calderon projector on the boundary and the projector defining the boundary condition. To make sense of this statement we need to develop a considerable extension of the classical notion of a Fredholm pair of subspaces of a Hilbert space. We call this the theory of "tame Fredholm pairs;" it is presented in the a…

On the singular Weinstein conjecture and the existence of escape orbits for $b$-Beltrami fields

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…

奇异Weinstein猜想

The singular Weinstein conjecture

本文研究$b^m$-切触流形上的Reeb动力学，推广Weinstein猜想，给出无周期轨道的紧流形例子，证明三维临界集有无穷多周期轨道，推广Hofer论证得到非紧流形的轨道存在性，并应用于三体问题。

The Arnold-Givental conjecture and moment Floer homology

We prove the Arnold-Givental conjecture for a class of Lagrangian submanifolds in Marsden-Weinstein quotients which are fixpoint sets of some antisymplectic involution. For these Lagrangians the Floer homology cannot in general be defined by standard means due to the bubbling phenomenon. To overcome this difficulty we consider moment Floer homology whose boundary operator is defined by counting solutions of the symplectic vortex equations on the strip which have better compactness properties than the original Floer equations.

A global Weinstein splitting theorem for holomorphic Poisson manifolds

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.

Characteristic classes of star products on Marsden-Weinstein reduced symplectic manifolds

In this note we consider a quantum reduction scheme in deformation quantization on symplectic manifolds proposed by Bordemann, Herbig and Waldmann based on BRST cohomology. We explicitly construct the induced map on equivalence classes of star products which will turn out to be an analogue to the Kirwan map but in the Cartan model of equivariant cohomology. As a byproduct we shall see that every star product on a (suitable) reduced manifold is equivalent to a reduced star product.

Contact topology and hydrodynamics II: solid tori

We prove the existence of periodic orbits for steady $C^ω$ Euler flows on all Riemannian solid tori. By using the correspondence theorem from part I of this series, we reduce the problem to the Weinstein Conjecture for solid tori. We prove the Weinstein Conjecture on the solid torus via a combination of results due to Hofer et al. and a careful analysis of tight contact structures on solid tori.

Linearization of proper groupoids

We prove the following result, conjectured by Alan Weinstein: every smooth proper Lie groupoid near a fixed point is locally linearizable, i.e. it is locally isomorphic to the associated groupoid of a linear action of a compact Lie group. In combination with a slice theorem of Weinstein, our result implies the smooth linearizability of a proper Lie groupoid in the neighborhood of an orbit under a mild condition.

lcs 流形中的椭圆曲线与度量不变量

Elliptic curves in lcs manifolds and metric invariants

本文在局部共形辛流形上通过带荷椭圆全纯曲线计数定义不变量，构造完备 Riemann-Finlser 流形的形变不变量，并推广 Weinstein 猜想至 lcs 几何，给出反例。

On the $\pd$- and $\barpd$-Operators of a Generalized Complex Structure

In this note, we prove that the $\pd$- and $\barpd$-operators introduced by Gualtieri for a generalized complex structure coincide with the $\bdees$- and $\bdel$-operators introduced by Alekseev-Xu for Evens-Lu-Weinstein modules of a Lie bialgebroid.

Two constructions of virtually contact structures

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.

Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids

In this note, we prove that intrinsic characteristic classes of Lie algebroids - which in degree one recover the modular class - behave functorially with respect to arbitrary transverse maps, and in particular are weak Morita invariants. In the modular case, this result appeared in [Kosmann-Schwarzbach Y., Laurent-Gengoux C., Weinstein A., Transform. Groups 13 (2008), 727-755], and with a connectivity assumption which we here show to be unnecessary, it appeared in [Crainic M., Comment. Math. Helv. 78 (2003), 681-721] and [Ginzburg V.L., J. Symplectic Geom. 1 (2001), 121-169].