The Seiberg–Witten equations and the Weinstein conjecture

Abstract

Let M denote a compact, oriented 3-dimensional manifold and let a denote a contact 1form on M; thus a ! da is nowhere zero. This article proves that the vector field that generates the kernel of da has a closed integral curve.

Sutures and contact homology I

“…Although ECH is defined in terms of a contact form, it is actually a topological invariant of the underlying 3-manifold, ie it does not depend on the contact structure (up to a possible grading shift -see Section 1.1). This invariance follows from a theorem of Taubes [37] identifying ECH with Seiberg-Witten Floer cohomology, which also implies the Weinstein conjecture in dimension three, again by work of Taubes [36].…”

On the Greenfield-Wallach and Katok conjectures in dimension three

“…The hard case which is left out at this point is the contact case. We can conclude the proof of the Katok conjecture by invoking the recent proof of the Weinstein conjecture by C. Taubes [Tau07]. In fact, by the Weinstein conjecture every Reeb flow in dimension 3 has at least a periodic orbit, hence cannot be uniquely ergodic, However, it seems important to develop different methods better adapted to our problem, especially in view of generalizations to higher dimensions.…”

“…The proof of the Greenfield-Wallach and Katok conjectures in dimension 3 is thus reduced to the proof of the Weinstein conjecture, recently announced by C. Taubes [Tau07]. However, it is important in our opinion to find an alternative proof in the contact case.…”

“…There are two cases: (a) ı X α ≡ 0 ; (b) ı X α ≡ 0; In case (a) it is possible to prove that M is a homogeneous manifolds and {φ X t } is a homogeneous flow, hence the Greenfield-Wallach Theorem 3.4 implies as above that M is a 3-torus, a contradiction. In case (b) the flow generated by X is the Reeb flow for the contact structure defined by the 1-form α, hence it has a periodic orbit by the Weinstein conjecture, recently proved by C. Taubes [Tau07]. However, every (CF) flow is volume preserving and uniquely ergodic, hence it cannot have periodic orbits.…”

“…The argument reduces the (CF) conjecture in the general case to the contact case which can be finished by invoking the proof of the Weinstein conjecture recently announced by C. Taubes [Tau07]. In fact, every (CF) flow is volume preserving and uniquely ergodic, while according to the Weinstein conjecture every contact flow on a closed, orientable 3-manifold has at least one periodic orbit.…”