Tag - Contact geometry

The question of whether a Symplectic manifold embeds into another is central in Symplectic topology. Since Gromov nonsqueezing theorem, it is known that this is a different problem from volume preserving embeddings. Symplectic capacities are invariants that give obstructions to symplectic embeddings. The first example of a symplectic capacity is given by the Gromov width, which measures the biggest ball that can be symplectically embedded into a symplectic manifold. In this talk, we are going to discuss the Gromov width for the example of disk cotangent bundles of spheres of revolution. The main results are for the Zoll cases and for the case of ellipsoids of revolution. The main tools are action angle coordinates (Arnold-Liouville theorem) and ECH capacities.

Contact homology is a Floer-type invariant for contact manifolds, and is a part of Symplectic Field Theory. One of its first applications was the existence of exotic contact structures on spheres. Originally, contact homology was defined only for closed contact manifolds. We will describe how to extend it to open contact manifolds that are 'convex'. As an application, we prove the existence of (infinitely many) exotic contact structures on ℝ²ⁿ⁺¹ for all n > 1.

Given a flow on a manifold, how to perturb it in order to create a periodic orbit passing through a given region? While the first results in this direction were obtained in the 1960s, various facets of this question remain largely open. I will review recent advances on this problem in the context of contact flows, which are closely related to Hamiltonian flows from classical mechanics. In particular, I'll discuss a proof of a conjecture of Irie stating that rotations of odd-dimensional ellipsoids admit a surprisingly large class of perturbations creating periodic orbits. The proof involves methods of modern symplectic topology including pseudo-holomorphic curves and contact homology.

In this talk, we start by reviewing recent results on the dynamics of Reeb vector fields defined by contact forms on three-dimensional manifolds, and then introduce Reeb fields defined by stable Hamiltonian structures. These are more general and arise, for instance, in stable regular energy level sets of Hamiltonian systems. We give a characterization of Reeb fields that are aperiodic or that have finitely many periodic orbits (under a certain nondegeneracy assumption). Finally, we give sufficient conditions for the existence of an adapted broken book decomposition or the existence of a Birkhoff section.

I will first give an overview of ECH. Then I will describe how to compute ECH in the Morse-Bott setting a la Bourgeois. I will discuss some classes of examples where this approach works. Finally I will sketch the gluing results that allow us to compute ECH using cascades.

ECH capacities have found many applications to symplectic embedding problems, most of which in the toric setting. I will discuss a new application of ECH to studying optimal embeddings for non-toric rational surfaces. The key convex geometric objects in our story are Newton-Okounkov bodies, which supply embeddings by work of Kaveh and also compute ECH capacities by joint recent work with Julian Chaidez.