Curves In Symplectizations Research Articles

The symplectic isotopy problem for rational cuspidal curves

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology.

Symmetric periodic orbits and invariant disk-like global surfaces of section on the three-sphere

We study Reeb dynamics on the three-sphere equipped with a tight contact form and an anti-contact involution. We prove the existence of a symmetric periodic orbit and provide necessary and sufficient conditions for it to bound an invariant disk-like global surface of section. We also study the same questions under the presence of additional symmetry and obtain similar results in this case. The proofs make use of pseudoholomorphic curves in symplectizations. As applications, we study Birkhoff’s conjecture on disk-like global surfaces of section in the planar circular restricted three-body problem and the existence of symmetric closed Finsler geodesics on the two-sphere. We also present applications to some classical Hamiltonian systems.

A compactness result for $\mathcal{H}$‑holomorphic curves in symplectizations

$\mathcal{H}-$holomorphic curves are solutions of a specific modification of the pseudoholomorphic curve equation in symplectizations involving a harmonic $1-$form as perturbation term. In this paper we compactify the moduli space of $\mathcal{H}-$holomorphic curves with a priori bounds on the harmonic $1-$forms.

The Characterization of Affine Symplectic Curves in ℝ4

Symplectic geometry arises as the natural geometry of phase-space in the equations of classical mechanics. In this study, we obtain new characterizations of regular symplectic curves with respect to the Frenet frame in four-dimensional symplectic space. We also give the characterizations of the symplectic circular helices as the third- and fourth-order differential equations involving the symplectic curvatures.

Periodic orbits in virtually contact structures

We prove that certain non-exact magnetic Hamiltonian systems on products of closed hyperbolic surfaces and with a potential function of large oscillation admit non-constant contractible periodic solutions of energy below the Mañé critical value. For that we develop a theory of holomorphic curves in symplectizations of non-compact contact manifolds that arise as the covering space of a virtually contact structure whose contact form is bounded with all derivatives up to order three.

Symplectic packings in dimension $4$ and singular curves

The main goal of this paper is to give constructive proofs of several existence results for symplectic embeddings. The strong relation between symplectic packings and singular symplectic curves, which can be derived from McDuff’s inflations on the blow-ups, is revisited through a new inflation technique that lives at the level of the manifold. As an application, we explain constructions of maximal symplectic packings of P by 6, 7 or 8 balls.

Global properties of tight Reeb flows with applications to Finsler geodesic flows on S2

AbstractWe show that if a Finsler metric on S2 with reversibility r has flag curvatures K satisfying (r/(r+1))2 < K ≤ 1, then closed geodesics with specific contact-topological properties cannot exist, in particular there are no closed geodesics with precisely one transverse self-intersection point. This is a special case of a more general phenomenon, and other closed geodesics with many self-intersections are also excluded. We provide examples of Randers type, obtained by suitably modifying the metrics constructed by Katok [21], proving that this pinching condition is sharp. Our methods are borrowed from the theory of pseudo-holomorphic curves in symplectizations. Finally, we study global dynamical aspects of 3-dimensional energy levels C2-close to S3

Gluing pseudoholomorphic curves along branched covered cylinders II

This paper and its prequel (“Part I”) prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves U+ and U− in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit γ, the total multiplicity of the negative ends of U+ at covers of γ agrees with the total multiplicity of the positive ends of U− at covers of γ. However, unlike in the usual gluing story, here the individual multiplicities are allowed to differ. In this situation, one can often glue U+ and U− to an index 2 curve by inserting genus zero branched covers of R-invariant cylinders between them. This paper shows that the signed count of such gluings equals a signed count of zeroes of a certain section of an obstruction bundle over the moduli space of branched covers of the cylinder. Part I obtained a combinatorial formula for the latter count and, assuming the result of the present paper, deduced that the differential ∂ in embedded contact homology satisfies ∂2 = 0. The present paper completes all of the analysis that was needed in Part I. The gluing technique explained here is in principle applicable to more gluing problems. We also prove some lemmas concerning the generic behavior of pseudoholomorphic curves in symplectizations, which may be of independent interest.

Simple Examples of Symplectic Four-manifolds with Exotic Properties

We construct examples of simply connected nonalgebraic symplectic four-manifolds with a prescribed number of nonintersecting symplectic curves with positive self-intersections.

An index inequality for embedded pseudoholomorphic curves in symplectizations

Let D be a surface with a symplectic form, let J be a symplectomorphism of D, and let Y be the mapping torus of J. We show that the dimensions of moduli spaces of embedded pseudoholomorphic curves in Â2Y, with cylindrical ends asymptotic to periodic orbits of J or multiple covers thereof, are bounded from above by an additive relative index. We deduce some compactness results for these moduli spaces. This paper establishes some of the foundations for a program with Michael Thaddeus, to understand the Seiberg-Witten Floer homology of Y in terms of such pseudoholomorphic curves. Analogues of our results should also hold in three dimensional contact topology.

Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three

l. In t roduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 1.1. The Weinstein conjecture . . . . . . . . . . . . . . . . . . . . . . . . 515 1.2. The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 1.3. Sketch of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 2. Local F redho lm theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 2.1. Similar i ty principle and consequences . . . . . . . . . . . . . . . . . . . 525 2.2. N_ • M as a lmost complex manifold and local fillings . . . . . . . . . . . . . 529 2.3. Local propert ies of ho lomorphic curves . . . . . . . . . . . . . . . . . . 531 3. Bubbl ing off analysis and global Fredholm theory . . . . . . . . . . . . . . . . 533 3.l. Grad ien t bounds imply Ca -bounds . . . . . . . . . . . . . . . . . . . . 533 3.2. Bubbing off analysis of ho lomorphic planes . . . . . . . . . . . . . . . . . 534 3.3. Grad ien t bounds near the boundary . . . . . . . . . . . . . . . . . . . . 537 4. Proof of the main results . . . . . . . . . . . . . . . . . . . . . . . . . . 544 4.1. Set-up for a global Fredholm theory . . . . . . . . . . . . . . . . . . . . 544 4.2. Bishop's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 4.3. Proof of theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 4.4. Proof of theorems 8 and 11 . . . . . . . . . . . . . . . . . . . . . . . 547 4.5. Uniqueness results for foliations . . . . . . . . . . . . . . . . . . . . . 547 5. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 5.1. Per turbing surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 5.2. Ki l l ing crit ical points of hypersurfaces . . . . . . . . . . . . . . . . . . . 552 5.3. Adding a thin tube to a sphere . . . . . . . . . . . . . . . . . . . . . . 560