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We relate the machinery of persistence modules to the Legendrian contact homology theory and to Poisson bracket invariants, and use it to show the existence of connecting trajectories of contact and symplectic Hamiltonian flows.
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An exact Lagrangian submanifold $L$ in the symplectization of standard contact $(2n-1)$-space with Legendrian boundary $Sigma$ can be glued to itself along $Sigma$. This gives a Legendrian embedding $Lambda(L,L)$ of the double of $L$ into contact $(2n+1)$-space. We show that the Legendrian isotopy class of $Lambda(L,L)$ is determined by formal data: the manifold $L$ together with a trivialization of its complexified tangent bundle. In particular, if $L$ is a disk then $Lambda(L,L)$ is the Legendrian unknot.
We present the Hamiltonian formalism for the Euler equation of symplectic fluids, introduce symplectic vorticity, and study related invariants. In particular, this allows one to extend D.Ebins long-time existence result for geodesics on the symplectomorphism group to metrics not necessarily compatible with the symplectic structure. We also study the dynamics of symplectic point vortices, describe their symmetry groups and integrability.
In this article, we prove a Legendrian Whitney trick which allows for the removal of intersections between codimension-two contact submanifolds and Legendrian submanifolds, assuming such a smooth cancellation is possible. This technique is applied to show the existence h-principle for codimension-two contact embeddings with a prescribed contact structure.
We show that there is no positive loop inside the component of a fiber in the space of Legendrian embeddings in the contact manifold $ST^*M$, provided that the universal cover of $M$ is $RM^n$. We consider some related results in the space of one-jets of functions on a compact manifold. We give an application to the positive isotopies in homogeneous neighborhoods of surfaces in a tight contact 3-manifold.
Let $X$ be a Weinstein manifold with ideal contact boundary $Y$. If $Lambdasubset Y$ is a link of Legendrian spheres in $Y$ then by attaching Weinstein handles to $X$ along $Lambda$ we get a Weinstein cobordism $X_{Lambda}$ with a collection of Lagrangian co-core disks $C$ corresponding to $Lambda$. In cite{BEE, EL} it was shown that the wrapped Floer cohomology $CW^{ast}(C)$ of $C$ in the Weinstein manifold $X_{Lambda}=Xcup X_{Lambda}$is naturally isomorphic to the Legendrian differential graded algebra $CE^{ast}(Lambda)$ of $Lambda$ in $Y$. The argument uses properties of moduli spaces of holomorphic curves, the proofs of which were only sketched. The purpose of this paper is to provide proofs of these properties.
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