Taubes’s proof of the Weinstein conjecture in dimension three

Abstract

Does every smooth vector field on a closed three-manifold, for example the three-sphere, have a closed orbit? No, according to counterexamples by K. Kuperberg and others. On the other hand there is a special class of vector fields, called Reeb vector fields, which are associated to contact forms. The three-dimensional case of the Weinstein conjecture asserts that every Reeb vector field on a closed oriented three-manifold has a closed orbit. This conjecture was recently proved by Taubes using Seiberg-Witten theory. We give an introduction to the Weinstein conjecture, the main ideas in Taubes’s proof, and the bigger picture into which it fits. Taubes’s proof of the Weinstein conjecture is the culmination of a large body of work, both by Taubes and by others. In an attempt to make this story accessible to nonspecialists, much of the present article is devoted to background and context, and Taubes’s proof itself is only partially explained. Hopefully this article will help prepare the reader to learn the full story from Taubes’s paper [62]. More exposition of this subject (which was invaluable in the preparation of this article) can be found in the online video archive from the June 2008 MSRI hot topics workshop [44], and in the article by Auroux [5]. Below, in §1–§3 we introduce the statement of the Weinstein conjecture and discuss some examples. In §4–§6 we discuss a natural strategy for approaching the Weinstein conjecture, which proves it in many but not all cases, and provides background for Taubes’s work. In §7 we give an overview of the big picture surrounding Taubes’s proof of the Weinstein conjecture. Readers who already have some familiarity with the Weinstein conjecture may wish to start here. In §8–§9 we recall necessary material from Seiberg-Witten theory. In §10 we give an outline of Taubes’s proof, and in §11 we explain some more details of it. To conclude, in §12 we discuss some further results and open problems related to the Weinstein conjecture. 1. Statement of the Weinstein conjecture The Weinstein conjecture asserts that certain vector fields must have closed orbits. Before stating the conjecture at the end of this section, we first outline its origins. This is discussion is only semi-historical, because only a sample of the relevant works will be cited, and not always in chronological order. 1.1. Closed orbits of vector fields. Let Y be a closed manifold (in this article all manifolds and all objects defined on them are smooth unless otherwise stated), 2000 Mathematics Subject Classification. 57R17,57R57,53D40. Partially supported by NSF grant DMS-0806037.