Exact Symplectic Manifold Research Articles

Nearly Periodic Maps and Geometric Integration of Noncanonical Hamiltonian Systems

M. Kruskal showed that each continuous-time nearly periodic dynamical system admits a formal U(1)-symmetry, generated by the so-called roto-rate. When the nearly periodic system is also Hamiltonian, Noether’s theorem implies the existence of a corresponding adiabatic invariant. We develop a discrete-time analog of Kruskal’s theory. Nearly periodic maps are defined as parameter-dependent diffeomorphisms that limit to rotations along a U(1)-action. When the limiting rotation is non-resonant, these maps admit formal U(1)-symmetries to all orders in perturbation theory. For Hamiltonian nearly periodic maps on exact presymplectic manifolds, we prove that the formal U(1)-symmetry gives rise to a discrete-time adiabatic invariant using a discrete-time extension of Noether’s theorem. When the unperturbed U(1)-orbits are contractible, we also find a discrete-time adiabatic invariant for mappings that are merely presymplectic, rather than Hamiltonian. As an application of the theory, we use it to develop a novel technique for geometric integration of non-canonical Hamiltonian systems on exact symplectic manifolds.

A Minimal Maslov Number Condition for Displaceability in Certain Weakly Exact Symplectic Manifolds

A Minimal Maslov Number Condition for Displaceability in Certain Weakly Exact Symplectic Manifolds

Noncommutative symplectic manifolds for self-injective Nakayama algebras

In the paper, we study noncommutative symplectic manifolds (A,llbracket omega rrbracket ), where A is a self-injective Nakayama algebra. There is given a full description of exact symplectic manifolds for A being the trivial extension of an hereditary Nakayama algebra.

A stable ∞-category of Lagrangian cobordisms

Given an exact symplectic manifold M and a support Lagrangian Λ⊂M, we construct an ∞-category LagΛ(M) which we conjecture to be equivalent (after specialization of the coefficients) to the partially wrapped Fukaya category of M relative to Λ. Roughly speaking, the objects of LagΛ(M) are Lagrangian branes inside of M×T⁎Rn, for large n, and the morphisms are Lagrangian cobordisms that are non-characteristic with respect to Λ. The main theorem of this paper is that LagΛ(M) is a stable ∞-category, and in particular its homotopy category is triangulated, with mapping cones given by an elementary construction. The shift functor is equivalent to the familiar shift of grading for Lagrangian branes.

Bounds for tentacular Hamiltonians

This paper represents a first step toward the extension of the definition of Rabinowitz Floer homology to non-compact energy hypersurfaces in exact symplectic manifolds. More concretely, we study under which conditions it is possible to establish [Formula: see text]-bounds for the Floer trajectories of a Hamiltonian with non-compact energy levels. Moreover, we introduce a class of Hamiltonians, called tentacular Hamiltonians which satisfy the conditions: how to define Rabinowitz Floer homology for these examples will be the subject of a follow-up paper.

Hamiltonian group actions on exact symplectic manifolds with proper momentum maps are standard

We give a complete characterization of Hamiltonian actions of compact Lie groups on exact symplectic manifolds with proper momentum maps. We deduce that every Hamiltonian action of a compact Lie group on a contractible symplectic manifold with a proper momentum map is globally linearizable.

Conley Conjecture Revisited

We show that whenever a closed symplectic manifold admits a Hamiltonian diffeomorphism with finitely many simple periodic orbits, the manifold has a spherical homology class of degree two with positive symplectic area and positive integral of the first Chern class. This theorem encompasses all known cases of the Conley conjecture (symplectic CY and negative monotone manifolds) and also some new ones (e.g., weakly exact symplectic manifolds with non-vanishing first Chern class). The proof hinges on a general Lusternik-Schnirelmann type result that, under some natural additional conditions, the sequence of mean spectral invariants for the iterations of a Hamiltonian diffeomorphism never stabilizes. We also show that for the iterations of a Hamiltonian diffeomorphism with finitely many periodic orbits the sequence of action gaps between the largest and the smallest spectral invariants remains bounded and, as consequence, establish some new cases of the $C^\infty$-generic existence of infinitely many simple periodic orbits.

$S^1$-equivariant Rabinowitz--Floer homology

We define the $S^1$-equivariant Rabinowitz--Floer homology of a bounding contact hypersurface $\Sigma$ in an exact symplectic manifold, and show by a geometric argument that it vanishes if $\Sigma$ is displaceable.

A Spectral Sequence of the Floer Cohomology of Symplectomorphisms of Trivial Polarization Class

Let $M$ be an exact symplectic manifold equal to a symplectization near infinity and having stably trivializable tangent bundle, and $\phi$ be an exact symplectomorphism of $M$ which, near infinity, is equal to either the identity or the symplectization of a contactomorphism $\hat{\phi}$ such that neither $\hat{\phi}$ nor $\hat{\phi}^2$ has fixed points. We give conditions under which Seidel and Smith's localization theorem for Lagrangian Floer cohomology implies the existence of a spectral sequence from $\mathit{HF}(\phi^2)\otimes \mathbb Z_2((\theta))$ to $\mathit{HF}(\phi)\otimes \mathbb Z_2((\theta))$.

Lifting pseudo-holomorphic polygons to the symplectisation of $P \times \mathbb{R}$ and applications

Let \mathbb R \times (P \times \mathbb R) be the symplectisation of the contactisation of an exact symplectic manifold P , and let \mathbb R \times \Lambda be a cylinder over a Legendrian submanifold of the contactisation. We show that a pseudo-holomorphic polygon in P having boundary on the projection of \Lambda can be lifted to a pseudo-holomorphic disc in the symplectisation having boundary on \mathbb R \times \Lambda . It follows that Legendrian contact homology may be equivalently defined by counting either of these objects. Using this result, we give a proof of Seidel's isomorphism of the linearised Legendrian contact homology induced by an exact Lagrangian filling and the singular homology of the filling.

Legendrian knots and exact Lagrangian cobordisms

We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair (X,L) consisting of an exact symplectic manifold X and an exact Lagrangian cobordism L\subset X which agrees with cylinders over Legendrian links \Lambda_+ and \Lambda_- at the positive and negative ends induces a differential graded algebra (DGA) map from the Legendrian contact homology DGA of \Lambda_+ to that of \Lambda_- . We give a gradient flow tree description of the DGA maps for certain pairs (X,L) , which in turn yields a purely combinatorial description of the cobordism map for elementary cobordisms, i.e., cobordisms that correspond to certain local modifications of Legendrian knots. As an application, we find exact Lagrangian surfaces that fill a fixed Legendrian link and are not isotopic through exact Lagrangian surfaces.

Floer cohomology of -equivariant Lagrangian branes

Building on Seidel and Solomon’s fundamental work [Symplectic cohomology and$q$-intersection numbers, Geom. Funct. Anal.22(2012), 443–477], we define the notion of a$\mathfrak{g}$-equivariant Lagrangian brane in an exact symplectic manifold$M$, where$\mathfrak{g}\subset SH^{1}(M)$is a sub-Lie algebra of the symplectic cohomology of$M$. When$M$is a (symplectic) mirror to an (algebraic) homogeneous space$G/P$, homological mirror symmetry predicts that there is an embedding of$\mathfrak{g}$in$SH^{1}(M)$. This allows us to study a mirror theory to classical constructions of Borel, Weil and Bott. We give explicit computations recovering all finite-dimensional irreducible representations of$\mathfrak{sl}_{2}$as representations on the Floer cohomology of an$\mathfrak{sl}_{2}$-equivariant Lagrangian brane and discuss generalizations to arbitrary finite-dimensional semisimple Lie algebras.

Picard–Lefschetz theory and dilating ℂ*-actions

We consider C^*-actions on Fukaya categories of exact symplectic manifolds. Such actions can be constructed by dimensional induction, going from the fibre of a Lefschetz fibration to its total space. We explore applications to the topology of Lagrangian submanifolds, with an emphasis on ease of computation.

A remark on unique ergodicity and the contact type condition

We prove that for a broad class of exact symplectic manifolds including \({\mathbb{R}^{2m}}\), the Hamiltonian flow on a regular compact energy level of an autonomous Hamiltonian cannot be uniquely ergodic. This is a consequence of the Weinstein conjecture and the observation that a Hamiltonian structure with non-vanishing self-linking number must have contact type. We apply these results to show that certain types of exact twisted geodesic flows cannot be uniquely ergodic.

A double Poisson algebra structure on Fukaya categories

Let $M$ be an exact symplectic manifold with $c_1(M)=0$. Denote by $\mathrm{Fuk}(M)$ the Fukaya category of $M$. We show that the dual space of the bar construction of $\mathrm{Fuk}(M)$ has a differential graded noncommutative Poisson structure. As a corollary we get a Lie algebra structure on the cyclic cohomology $\mathrm{HC}^\bullet(\mathrm{Fuk}(M))$, which is analogous to the ones discovered by Kontsevich in noncommutative symplectic geometry and by Chas and Sullivan in string topology.

On the Strict Arnold Chord Property and Coisotropic Submanifolds of Complex Projective Space

Let be a contact form on a manifoldM, and L M a closed Legendrian submanifold. I prove that L intersects some characteristic for at least twice if all characteristics are closed and of the same period, and embeds nicely into the product of R2n and an exact symplectic manifold. As an application of the method of proof, the minimal action of a regular closed coisotropic submanifold of complex projective space is at most /2. This yields an obstruction to presymplectic embeddings, and in particular to Lagrangian embeddings.

Lie bialgebra structure on cyclic cohomology of Fukaya categories

Let M be an exact symplectic manifold with contact type boundary such that c1(M) = 0. Motivated by noncommutative symplectic geometry and string topology, we show that the cyclic cohomology of the Fukaya category of M has an involutive Lie bialgebra structure.

Symplectic A-directed immersions

We prove a symplectic analogue of A-directed immersion theorem for exact symplectic manifolds and generalize this result to closed manifolds under a completeness hypothesis. As an application we have derived a sufficient condition for the existence of a Poisson structure on an exact symplectic manifold with a codimension-2q characteristic foliation such that the Poisson structure is induced from the ambient symplectic structure.

When are the invariant submanifolds of symplectic dynamics Lagrangian?

Let $\mathcal{L}$ be a $D$-dimensional submanifold of a $2D$ dimensional exact symplectic manifold $(M, \omega)$ and let $f: M\rightarrow M$ be a symplectic diffeomorphism. In this article, we deal with the link between the dynamics $f_{|\mathcal{L}}$ restricted to $\mathcal{L}$ and the geometry of $\mathcal{L}$ (is $\mathcal{L}$ Lagrangian, is it smooth, is it a graph … ?). We prove different kinds of results. 1. for $D=3$, we prove that is $\mathcal{L}$ if a torus that carries some characteristic loop, then either $\mathcal{L}$ is Lagrangian or $f_{|\mathcal{L}}$ can not be minimal (i.e. all the orbits are dense) with $(f^k_{|\mathcal{L}})$ equilipschitz; 2. for a Tonelli Hamiltonian of $T^*\mathbb{T}^3$, we give an example of an invariant submanifold $\mathcal{L}$ with no conjugate points that is not Lagrangian and such that for every $f:T^*\mathbb{T}^3\rightarrow T^*\mathbb{T}^3$ symplectic, if $f(\mathcal{L})=\mathcal{L}$, then $\mathcal{L}$ is not minimal; 3. with some hypothesis for the restricted dynamics, we prove that some invariant Lipschitz $D$-dimensional submanifolds of Tonelli Hamiltonian flows are in fact Lagrangian, $C^1$ and graphs; 4.we give similar results for $C^1$ submanifolds with weaker dynamical assumptions.

Dehn twists and free subgroups of symplectic mapping class groups

Given two Lagrangian spheres in an exact symplectic manifold, we find conditions under which the Dehn twists about them generate a free non-abelian subgroup of the symplectic mapping class group. This extends a result of Ishida for Riemann surfaces. The proof generalises the categorical version of Seidel's long exact sequence to arbitrary powers of a fixed Dehn twist. We also show that the Milnor fibre of any isolated degenerate hypersurface singularity contains such pairs of spheres.