On the Weinstein Conjecture in Higher Dimensions

The first break-through on this conjecture was obtained by C. Viterbo, [19], showing that compact energy surfaces in R2n of contact-type have periodic orbits. Extending Gromov’s theory of pseudoholomorphic curves, [3], to symplectized contact manifolds, H. Hofer, [4], related the Weinstein conjecture to the existence of certain pseudoholomorphic curves. He showed that in dimension three the Weinstein conjectures holds in many cases. In particular, he showed that Reeb vector fields associated to over-twisted contact structures admit periodic orbits. Recently the Weinstein conjecture in dimension three was completely settled by C. Taubes, [17, 18], who exploited relationships between Seiberg-Witten-Floer homology, [12], and embedded contact homology, [11], in order to construct holomorphic curves in the symplectized contact manifold out of nontrivial Seiberg-Witten-Floer homology classes. For more references on the Weinstein conjecture see [6]. In this note we show that many Reeb vector fields on higher dimensional closed manifolds have periodic orbits generalizing the main result from [4]. Our existence result is closely connected to the interesting attempt by K. Niederkrüger [13] to generalize the three-dimensional notion of an overtwisted contact structure. He introduced the concept of a Plastikstufe which currently seems to be the most compelling generalisation given recent further developments by F. Presas, [15] and K. Niederkrüger / O. van Koert, [14]. Let us denote by (M, ξ) a pair consisting of a closed manifold M of dimension 2n − 1 and a co-oriented contact structure ξ. We denote by D2 the closed unit disk in C with coordinates x + iy.