Taubes’s proof of the Weinstein conjecture in dimension three

Hutchings, Michael

doi:10.1090/S0273-0979-09-01282-8

Taubes’s proof of the Weinstein conjecture in dimension three
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Bull. Amer. Math. Soc. 47 (2010), 73-125 Request permission

Abstract:

Does every smooth vector field on a closed three-manifold, for example the three-sphere, have a closed orbit? The answer is 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.

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