In this paper we describe a general strategy for approaching the Weinstein conjecture in dimension three. We apply this approach to prove the Weinstein conjecture for a new class of contact manifolds (planar contact manifolds). We also discuss how the present approach reduces the general Weinstein conjecture in dimension three to a compactness problem for the solution set of a first order elliptic PDE.

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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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