Floer's chain complexes for Lagrangian submanifolds in symplectic manifolds with concave ends

拉格朗日配边的Floer理论

Floer theory for Lagrangian cobordisms

本文在Liouville流形接触化中为恰当拉格朗日配边建立交Floer同调，并借助正合序列研究其拓扑性质。

Unwrapped continuation invariance in Lagrangian Floer theory: energy and $C^0$ estimates

We consider pairs of Lagrangian submanifolds $(L_0,L), (L_1, L)$ belonging to the class of Lagrangian submanifolds with \emph{conic} ends on \emph{Weinstein manifolds}. The main purpose of the present paper is to define a canonical chain map $h_\CL: CF(L_0,L) \to CF(L_1,L)$ of Lagrangian Floer complex inducing an isomorphism in homology, under the Hamiltonian isotopy $\CL=\{L_s\}_{0 \leq s\leq 1}$ generated by \emph{conic} Hamiltonian functions such that the intersections $L \cap L_s$ do not escape to infinity. The main ingredients of the proof is an a priori bound for general isotopy of the e…

Chain level Floer theory and Hofer's geometry of the Hamiltonian diffeomorphism group

In this paper we first apply the chain level Floer theory to the study of Hofer's geometry of Hamiltonian diffeomorphism group in the cases without quantum contribution: we prove that any quasi-autonomous Hamiltonian path on weakly exact symplectic manifolds or any autonomous Hamiltonian path on arbitrary symplectic manifolds is length minimizing in its homotopy class with fixed ends, as long as it has a fixed maximum and a fixed minimum which are not over-twisted and all of its contractible periodic orbits of period less than one are sufficiently $C^1$-small. Next we give a construction of ne…

A Hardy Space Approach to Lagrangian Floer Gluing

We develop a new approach to Lagrangian-Floer gluing. The construction of the gluing map is based on the intersection theory in some Hilbert manifold of paths $\mathcal{P} $. We consider some moduli spaces of perturbed holomorphic curves whose domains are either strips or more general Riemann surfaces with strip-like ends. These moduli spaces can be injectively immersed into the Hilbert manifold $\mathcal{P} $ by taking the restriction to non-Lagrangian boundary. Some subsets $ \mathcal{M}^T(U) $ and $ \mathcal{M}^{\infty} (U) $ of the aforementioned moduli spaces of perturbed holomorphic stri…

Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology

We relate the version of rational Symplectic Field Theory for exact Lagrangian cobordisms introduced in [5] with linearized Legendrian contact homology. More precisely, if $L\subset X$ is an exact Lagrangian submanifold of an exact symplectic manifold with convex end $Λ\subset Y$, where $Y$ is a contact manifold and $Λ$ is a Legendrian submanifold, and if $L$ has empty concave end, then the linearized Legendrian contact cohomology of $Λ$, linearized with respect to the augmentation induced by $L$, equals the rational SFT of $(X,L)$. Following ideas of P. Seidel, this equality in combination wi…

Product structures in Floer theory for Lagrangian cobordisms

We construct a product on the Floer complex associated to a pair of Lagrangian cobordisms. More precisely, given three exact transverse Lagrangian cobordisms in the symplectization of a contact manifold, we define a map $\mathfrak{m}_2$ by a count of rigid pseudo-holomorphic disks with boundary on the cobordisms and having punctures asymptotic to intersection points and Reeb chords of the negative Legendrian ends of the cobordisms. More generally, to a $(d+1)$-tuple of exact transverse Lagrangian cobordisms we associate a map $\mathfrak{m}_d$ such that the family $(\mathfrak{m}_d)_{d\geq1}$ ar…