The Weinstein conjecture in the uniruled manifolds

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。

Pseudo-holomorphic curves and the Weinstein conjecture

We established existence of periodic Reeb orbits for a large class of tight contact structures on closed 3-manifolds, notably the Stein fillable structures, based on a fundamental theorem of Cliff Taubes on symplectic 4-manifolds.

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。

Finiteness of the Hofer-Zehnder capacity of neighborhoods of symplectic submanifolds

We use the minimal coupling procedure of Sternberg and Weinstein and our pseudo-symplectic capacity theory to prove that every closed symplectic submanifold in any symplectic manifold has an open neighborhood with finite ($π_1$-sensitive) Hofer-Zehnder symplectic capacity. Consequently, the Weinstein conjecture holds near closed symplectic submanifolds in any symplectic manifold.

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.

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。

Weinstein流形与扇形的缠绕Fukaya范畴的几何生成

Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors

该研究通过几何方法证明2n维Weinstein流形/扇形的缠绕Fukaya范畴由Liouville场n阶临界点不稳定流形生成，证得Seidel关于开闭映射为同构的猜想，并扩展了证明设定。

Affine manifolds, lagrangian manifolds

Let $(M,ω)$ be a symplectic manifold endowed with a agrangian foliation ${\cal L}$, it has been shown by Weinstein [16] hat the symplectic structure of $M$ defines on each leaf of ${\cal L}$, connection which curvature and torsion forms vanish identically. uppose that $L_0$ is a compact leaf which Weinstein connection is eodesically complete, Molino and Curras-Bosch [2] have classified erms of such lagrangian foliation around $L_0$. In this paper we xtend this classification without supposing the completness of he compact leaf. The Weinstein connection is dual to the Bott onnection, this enabl…

Gromov-Witten theory of a locally conformally symplectic manifold

We initiate here the study of Gromov-Witten theory of locally conformally symplectic manifolds or $\lcs$ manifolds, $\lcsm$'s for short, which are a natural generalization of both contact and symplectic manifolds. We find that the main new phenomenon (relative to the symplectic case) is the potential existence of holomorphic sky catastrophes, an analogue for pseudo-holomorphic curves of sky catastrophes in dynamical systems originally discovered by Fuller. We are able to rule these out in some situations, particularly for certain $\lcs$ 4-folds, and as one application we show that in dimension…

A Linking/$S^1$ Equivariant Variational Argument in the space of Dual Legendrian curves and the proof of the Weinstein Conjecture on $S^3$ "in the large"

Let $α$ be a contact form on $S^3$, let $ξ$ be its Reeb vector-field and let $v$ be a non-singular vector-field in $kerα$. Let $C_β$ be the space of curves $x$ on $S^3$ such $\dot x=aξ+bv, \dot a=0, a \gneq 0$. Let $L^+$, respectively $L^-$, be the set of curves in $C_β$ such that $b\geq 0$, respectively $b \leq 0$. Let, for $x \in C_β$, $J(x)=\int_0^1α_x(\dot x)dt$. We establish in this paper that an infinite number of cycles in the $S^1$-equivariant homology of $C_β$,{\bf relative} to $L^+ \cup L^-$ and to some specially designed "bottom set", see section 4, are achieved in the Morse complex…

Embedded H-holomorphic maps and open book decompositions

We investigate nicely embedded H--holomorphic maps into stable Hamiltonian three--manifolds. In particular we prove that such maps locally foliate and satisfy a no--first--intersection property. Using the compactness results of arXiv:0904.1603 we show that connected components of the space of such maps can be compactified if they contain a global surface of section. As an application we prove that any contact structure on a 3--manifold admits and H--holomorphic open book decomposition. This work is motivated by the program laid out by Abbas, Cieliebak and Hofer to give a proof to the Weinstein…

Geometrie affine Geometrie symplectique

Let (M,w,L) be a symplectic manifold endowed with a lagrangian foliation L. Liberman and Weinstein have shown that the leaves of L are endowed with an affine structure. In this paper we provide links between the theories of affine manifolds and symplectic geometry. Using the work of Donaldson who have shown the existence of symplectic submanifolds in every codimension, we show the Auslander conjecture to be true if the linear holonomy is contained in Gl(n,Z)

J-holomorphic Curves And Periodic Reeb Orbits

We study the $J-$holomorphic curves in the symplectization of the contact manifolds and prove that there exists at least one periodic Reeb orbits in any closed contact manifold with any contact form by using the well-known Gromov's nonlinear Fredholm alternative for $J-$holomorphic curves. As a corollary, we give a complete solution on the well-known Weinstein conjecture.

Floer homologies, with applications

Floer invented his theory in the mid eighties in order to prove the Arnol'd conjectures on the number of fixed point of Hamiltonian diffeomorphisms and Lagrangian intersections. Over the last thirty years, many versions of Floer homology have been constructed. In symplectic and contact dynamics and geometry they have become a principal tool, with applications that go far beyond the Arnol'd conjectures: The proof of the Conley conjecture and of many instances of the Weinstein conjecture, rigidity results on Lagrangian submanifolds and on the group of symplectomorphisms, lower bounds for the top…

Compactness results for H-holomorphic maps

H-holomorphic maps are a parameter version of J-holomorphic maps into contact manifolds. They have arisen in efforts to prove the existence of higher--genus holomorphic open book decompositions and efforts to prove the existence of finite energy foliations and the Weinstein conjecture, as well as in folded holomorphic maps. For all these applications it is essential to understand the compactness properties of the space of H-holomorphic maps. We prove that the space of H-holomorphic maps with bounded periods into a manifold with stable Hamiltonian structure possesses a natural compactification.…