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Register a new userAn energy-capacity inequality for Legendrian submanifolds
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Georgios Dimitroglou Rizell
Publication date
2016
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English
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We prove that the number of Reeb chords between a Legendrian submanifold and its contact Hamiltonian push-off is at least the sum of the $mathbb{Z}_2$-Betti numbers of the submanifold, provided that the contact isotopy is sufficiently small when compared to the smallest Reeb chord on the Legendrian. Moreover, the established invariance enables us to use two different contact forms: one for the count of Reeb chords and another for the measure of the smallest length, under the assumption that there is a suitable symplectic cobordism from the latter to the former. The size of the contact isotopy is measured in terms of the oscillation of the contact Hamiltonian, together with the maximal factor by which the contact form is shrunk during the isotopy. The main tool used is a Mayer--Vietoris sequence for Lagrangian Floer homology, obtained by neck-stretching and splashing.
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In any contact manifold of dimension $2n-1geq 11$, we construct examples of closed legendrian submanifolds which are not diffeomorphic but whose lagrangian cylinders in the symplectization are hamiltonian isotopic.
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.
Previously, Cristofaro-Gardiner, Hutchings and Ramos have proved that embedded contact homology (ECH) capacities can recover the volume of a contact 3-manifod in their paper the asymptotics of ECH capacities . There were two main steps to proving this theorem: The first step used an estimate for the energy of min-max Seiberg-Witten Floer generators. The second step used embedded balls in a certain symplectic four manifold. In this paper, stronger estimates on the energy of min-max Seiberg-Witten Floer generators are derived. This stronger estimate implies directly the ECH capacities recover volume theorem (without the help of embedded balls in a certain symplectic four manifold), and moreover, gives an estimate on its speed.
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.
In this article we study Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. First we provide a systematic recipe for translating from a Weinstein Lefschetz bifibration to a Legendrian handlebody. Then we present several applications of this technique to symplectic topology. This includes the detection of flexibility and rigidity for several families of Weinstein manifolds and the existence of closed exact Lagrangian submanifolds. In addition, we prove that the Koras--Russell cubic is Stein deformation equivalent to affine complex 3-space and verify the affine parts of the algebraic mirrors of two Weinstein 4-manifolds.
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