The Seiberg–Witten equations and the Weinstein conjecture

Let M denote a compact, orientable 3–manifold and let a denote a smooth 1–form on M such that a^ da is nowhere zero. Such a 1–form is called a contact form. The associated Reeb vector field is the section, v , of TM that generates the kernel of da and pairs with a to give 1. The generalized Weinstein conjecture in dimension three asserts that v has at least one closed integral curve (see Weinstein [30]). The purpose of this article is to prove this conjecture and somewhat more. To state the result, remark that the kernel of the 1–form a defines an oriented 2–plane subbundle K 1 TM . Since an oriented 3–manifold is spin, the first Chern class of this two-plane bundle is divisible by 2.