Four-dimensional Symplectic Manifolds Research Articles

Symplectic structures with non-isomorphic primitive cohomology on open 4-manifolds

We analyze four-dimensional symplectic manifolds of type X = S 1 × M 3 X=S^1 \times M^3 where M 3 M^3 is an open 3 3 -manifold admitting inequivalent fibrations leading to inequivalent symplectic structures on X X . For the case where M 3 ⊂ S 3 M^3 \subset S^3 is the complement of a 4 4 -component link constructed by McMullen-Taubes, we provide a general algorithm for computing the monodromy of the fibrations explicitly. We use this algorithm to show that certain inequivalent symplectic structures are distinguished by the dimensions of the primitive cohomologies of differential forms on X X . We also calculate the primitive cohomologies on X X for a class of open 3 3 -manifolds that are complements of a family of fibered graph links in S 3 S^3 . In this case, we show that there exist pairs of symplectic forms on X X , arising from either equivalent or inequivalent pairs of fibrations on the link complement, that have different dimensions of the primitive cohomologies.

Semi-Classical Discretization and Long-Time Evolution of Variable Spin Systems.

We apply the semi-classical limit of the generalized map for representation of variable-spin systems in a four-dimensional symplectic manifold and approximate their evolution terms of effective classical dynamics on . Using the asymptotic form of the star-product, we manage to “quantize” one of the classical dynamic variables and introduce a discretized version of the Truncated Wigner Approximation (TWA). Two emblematic examples of quantum dynamics (rotor in an external field and two coupled spins) are analyzed, and the results of exact, continuous, and discretized versions of TWA are compared.

Adiabatic limit in Ginzburg—Landau and Seiberg—Witten equations

We present the concept of an adiabatic limit of Ginzburg—Landau dynamical equations on ℝ1+2 and Seiberg—Witten equations on four-dimensional symplectic manifolds. We show that the Seiberg—Witten equations can be regarded as a complex version of the Ginzburg—Landau equations.

Beyond ECH capacities

ECH (embedded contact homology) capacities give obstructions to symplectically embedding one four-dimensional symplectic manifold with boundary into another. These obstructions are known to be sharp when the domain and target are ellipsoids (proved by McDuff), and more generally when the domain is a "concave toric domain" and the target is a "convex toric domain" (proved by Cristofaro-Gardiner). However ECH capacities often do not give sharp obstructions, for example in many cases when the domain is a polydisk. This paper uses more refined information from ECH to give stronger symplectic embedding obstructions when the domain is a polydisk, or more generally a convex toric domain. We use these new obstructions to reprove a result of Hind-Lisi on symplectic embeddings of a polydisk into a ball, and generalize this to obstruct some symplectic embeddings of a polydisk into an ellipsoid. We also obtain a new obstruction to symplectically embedding one polydisk into another, in particular proving the four-dimensional case of a conjecture of Schlenk.

Semiclassical phase-space dynamics of compound quantum systems: SU(2) covariant approach

The semiclassical evolution of the generalized SU(2) Wigner function for quantum systems with a variable number of excitations is discussed in detail. It is shown that in the framework of the Liouvillian approach quantum dynamics can be approximately described in terms of classical trajectories on an appropriate four-dimensional symplectic manifold. The analysis of this manifold allows the introduction of a phase-like observable that can be interpreted as a relative phase ‘between’ invariant subspaces.

The asymptotics of ECH capacities

In a previous paper, the second author used embedded contact homology (ECH) of contact three-manifolds to define “ECH capacities” of four-dimensional symplectic manifolds. In the present paper we prove that for a four-dimensional Liouville domain with all ECH capacities finite, the asymptotics of the ECH capacities recover the symplectic volume. This follows from a more general theorem relating the volume of a contact three-manifold to the asymptotics of the amount of symplectic action needed to represent certain classes in ECH. The latter theorem was used by the first and second authors to show that every contact form on a closed three-manifold has at least two embedded Reeb orbits.

Singular Polarizations and Ellipsoid Packings

We prove in this paper that any four-dimensional symplectic manifold is essentially made of finitely many symplectic ellipsoids. The key tool is a singular analog of Donaldson’s symplectic hypersurfaces in irrational symplectic manifolds.

Recent progress on symplectic embedding problems in four dimensions

We survey some recent progress on understanding when one four-dimensional symplectic manifold can be symplectically embedded into another. In 2010, McDuff established a number-theoretic criterion for the existence of a symplectic embedding of one four-dimensional ellipsoid into another. This result is related to previously known criteria for when a disjoint union of balls can be symplectically embedded into a ball. Numerical invariants defined using embedded contact homology give general obstructions to symplectic embeddings in four dimensions which turn out to be sharp in the above cases.

Geometric Averaging of Hamiltonian Systems: Periodic Solutions, Stability, and KAM Tori

We investigate the dynamics of various problems defined by Hamiltonian systems of two and three degrees of freedom that have in common that they can be reduced by an axial symmetry. Specifically, the systems are either invariant under rotation about the vertical axis or can be made approximately axially symmetric after an averaging process and the corresponding truncation of higher-order terms. Once the systems are reduced we study the existence and stability of relative equilibria on the reduced spaces which are unbounded two- or four-dimensional symplectic manifolds with singular points. We establish the connections between the existence and stability of relative equilibria and the existence and stability of families of periodic solutions of the full problem. We also discuss the existence of KAM tori surrounding the periodic solutions.

On the stability of the set of hyperbolic closed orbits of a Hamiltonian

AbstractLetHbe a Hamiltonian,e∈H(M) ⊂ ℝ andƐH, ea connected component ofH−1({e}) without singularities. A Hamiltonian system, say a triple (H,e,ƐH, e), is Anosov ifƐH, eis uniformly hyperbolic. The Hamiltonian system (H,e,ƐH, e) is aHamiltonian star systemif all the closed orbits ofƐH, eare hyperbolic and the same holds for a connected component of−1({ẽ}), close toƐH, e, for any Hamiltonian, in someC2-neighbourhood ofH, and ẽ in some neighbourhood ofe.In this paper we show that a Hamiltonian star system, defined on a four-dimensional symplectic manifold, is Anosov. We also prove the stability conjecture for Hamiltonian systems on a four-dimensional symplectic manifold. Moreover, we prove the openness and the structural stability of Anosov Hamiltonian systems defined on a 2d-dimensional manifold,d≥ 2.

Monodromy Invariants – From Symplectic to Smooth Manifolds

Recently, together with Auroux and Donaldson, we have introduced some new invariants of four-dimensional symplectic manifolds. Building on the Moishezon–Teicher braid factorization techniques, we show how to compute fundamental groups of compliments to a ramification curve of generic projection. We also show that these fundamental groups are only homology invariants and outline the computations in some examples.

On semi-global invariants for focus–focus singularities

This article gives a classification, up to symplectic equivalence, of singular Lagrangian foliations given by a completely integrable system of a four-dimensional symplectic manifold, in a full neighbourhood of a singular leaf of focus–focus type.

The generic symplectic C^{1}-diffeomorphisms of four-dimensional symplectic manifolds are hyperbolic, partially hyperbolic or have a completely elliptic periodic point

We prove that if (M,\omega) is a connected and compact four-dimensional symplectic manifold, there exist three open sets U_1, U_2, U_3 of {\rm Diff}^1_{\omega}(M) (for the C^1 topology) such that: U_1\cup U_2\cup U_3 is dense in {\rm Diff}^1_{\omega}(M);f\in U_1 if and only if f is Anosov and transitive;f\in U_2 if and only if f is partially hyperbolic; andf\in U_3 if and only if f has a stable completely elliptic periodic point.

Typical integrable Hamiltonian systems on a four-dimensional symplectic manifold

We study the topology of integrable Hamiltonian systems with two degrees of freedom in the neighbourhood of a degenerate circle. Among all degenerate circles, the class of so-called generic degenerate circles is singled out. These circles cannot be removed from the symplectic manifold by a small perturbation of the Poisson action, and the system remains topologically equivalent to the unperturbed system in their neighbourhood. Moreover, if the unperturbed system has only Bott circles and generic degenerate circles, then, under the condition of simplicity, the perturbed system is globally topologically equivalent to it. It is proved that if an additional condition holds, then there is a small perturbation for which all degenerate circles are generic.

Classification of four-dimensional integrable Hamiltonian systems and Poisson actions of in extended neighbourhoods of simple singular points. III. Realization

The classification of integrable Hamiltonian systems in extended neighbourhoods of simple singular points, initiated in [1] and [2], is completed. This paper is a direct continuation of [1] and [2] and uses all the concepts, definitions, and results presented there. As promised in [2], the aim here is to realize all admissible types of the invariants describing the iso-energetic equivalence of integrable Hamiltonian systems with two degrees of freedom in extended neighbourhoods of simple singular points. In doing that, all admissible types of Poisson actions of the group on a four-dimensional symplectic manifold are also realized. Some of the examples constructed are of a purely illustrative nature, while others are more meaningful and occur in applications.

CLASSIFICATION OF FOUR-DIMENSIONAL INTEGRABLE HAMILTONIAN SYSTEMS AND POISSON ACTIONS OF IN EXTENDED NEIGHBORHOODS OF SIMPLE SINGULAR POINTS. I

Integrable Hamiltonian systems and Poisson actions of the group with simple singular points on a smooth ( or real-analytic) four-dimensional symplectic manifold are studied, where is a symplectic 2-form. Bibliography: 18 titles.

CLASSIFICATION OF FOUR-DIMENSIONAL INTEGRABLE HAMILTONIAN SYSTEMS AND POISSON ACTIONS OF IN EXTENDED NEIGHBORHOODS OF SIMPLE SINGULAR POINTS. II

Integrable Hamiltonian systems and Poisson actions of the group with simple singular points on a smooth ( or real-analytic) four-dimensional symplectic manifold are studied, where is a symplectic 2-form. Bibliography: 6 titles.

A topological invariant which roughly classifies integrable strictly nondegenerate Hamiltonians on four-dimensional symplectic manifolds

A topological invariant which roughly classifies integrable strictly nondegenerate Hamiltonians on four-dimensional symplectic manifolds

Entropy and Completely Integrable Hamiltonian Systems

Let $H$ be a Hamiltonian on a four-dimensional symplectic manifold. Suppose the system is completely integrable and on some nonsingular compact level surface $Q$ the integral is such that the connected components of the set of critical points form submanifolds. Then we prove that the topological entropy of the system restricted to $Q$ is zero. As a corollary we deduce the nonexistence of completely integrable geodesic flows by means of integrals as described above for compact surfaces with negative Euler characteristic.