Pseudoholomorphic strips in symplectizations II: Fredholm theory and transversality

Abstract

This paper is part of a larger program, the investigation of the chord problem in three-dimensional contact geometry. The main tool will be pseudoholomorphic strips in the symplectization of a three-dimensional contact manifold with two totally real submanifolds L 0 and L 1 as boundary conditions. The submanifolds L 0 and L 1 do not intersect transversally. In this paper we will develop a nonlinear Fredholm theory that guarantees the existence of a family of embedded pseudoholomorphic strips near a given one with suitable properties.

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“…A. Floer's proof of transversality by choosing a generic almost complex structure does not work here. With a bit more effort (using also that R × M is four dimensional) one can show that transversality holds automatically (see [5]). The previous papers [4], [5] and [6] take care of all these analytic issues.…”

“…The Maslov-index µ(ũ τ ) is a topological quantity associated to the boundary condition {0} × D. Its precise definition is not relevant for this paper, see [5].…”

“…The assumption (3) is some kind of geometric 'smallness' assumption on the Legendrian knot. The purpose of this paper is to give a first application of the filling method developed in [4], [5] and [6] without any further technical complications. The assumption inf(λ) > vol λ (D) is vacuous if there are no contractible periodic orbits of the Reeb vector field.…”

“…Dealing with these issues significantly complicates matters because families of pseudoholomorphic strips cannot be extended further once they hit such a singular point. In fact, the boundary value problem discussed in this and the previous papers [4], [5] and [6] would have to be modified in order to deal with these difficulties. These issues will be addressed in the forthcoming papers [7], [8].…”

“…A new invariant for Legendrian knots and contact manifolds proposed by Y. Eliashberg, A. Givental and H. Hofer in [16] ('Relative Contact Homology') is actually based on counting characteristic chords and periodic orbits of the Reeb vector field. In this paper we will use the method of filling by pseudoholomorphic curves which was developed in the previous papers [4], [5] and [6] in order to prove a first existence result for characteristic chords on more general contact three manifolds. Before we can state the main result of this paper we need a few more definitions.…”