K. Wysocki

Compactness results in Symplectic Field Theory

This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in (4). We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic Field Theory. The theorems generalize Gromovs compactness theorem in (8) as well as compactness theorems in Floer homology theory, (6, 7), and in contact geometry, (9, 19).

The dynamics on three-dimensional strictly convex energy surfaces

We show that a Hamiltonian flow on a three-dimensional strictly convex energy surface S C R4 possesses a global surface of section of disc type. It follows, in particular, that the number of its periodic orbits is either 2 or oc, by a recent result of J. Franks on area-preserving homeomorphisms of an open annulus in the plane. The construction of this surface of section is based on partial differential equations of Cauchy-Riemann type for maps from punctured Riemann surfaces into R x S3 equipped with special almost complex structures.

Properties of pseudo-holomorphic curves in symplectisations II: Embedding controls and algebraic invariants

In the following we look for conditions on a finite energy plane ũ : = (a, u) : ℂ → ℝ × M, which allow us to conclude that the projection into the manifold M, u : ℂ → M, is an embedding. For this purpose we shall introduce several algebraic invariants. Finite energy planes have been introduced in [H] for the solution of A. Weinstein’s conjecture about closed characteristics on three dimensional contact manifolds. In order to recall the concept, we first start with some definitions from contact geometry.

Properties of Pseudoholomorphic Curves in Symplectizations III: Fredholm Theory

We shall study smooth maps ũ: S → ℝ x M of finite energy defined on the punctured Riemann surface S = S\Γ and satisfying a Cauchy-Riemann type equation Tũ ∘ j = Jũ ∘ Tũ for special almost complex structures J, related to contact forms A on the compact three manifold M. Neither the domain nor the target space are compact. This difficulty leads to an asymptotic analysis near the punctures. A Fredholm theory determines the dimension of the solution space in terms of the asymptotic data defined by non-degenerate periodic solutions of the Reeb vector field associated with λ on M, the Euler characteristic of S, and the number of punctures. Furthermore, some transversality results are established.

UNKNOTTED PERIODIC ORBITS FOR REEB FLOWS ON THE THREE-SPHERE

It is well known that a Reeb vector field on

A characterization of the tight 3-sphere II

S^3

Finite energy cylinders of small area

has a periodic solution. Sharpening this result we shall show in this note that every Reeb vector field

BRANCH STRUCTURE OF J-HOLOMORPHIC CURVES NEAR PERIODIC ORBITS OF A CONTACT MANIFOLD

X

Chapter 15 Pseudoholomorphic curves and dynamics in three dimensions

on

A General Fredholm Theory II: Implicit Function Theorems

S^3