The Seiberg-Witten equations and the Weinstein conjecture. (English) Zbl 1135.57015

Let \(\mathcal{M}\) be a compact, orientable 3-dimensional manifold and let \(\alpha\) be a contact form. The kernel of the 1-form \(\alpha\) yields an oriented 2-plane subbundle \(K^{-1}\) of \(TM\). Weinstein conjectured that the associated Reeb vector field has at least one closed integral curve. The author proves this conjecture. Namely, for a fixed 2-dimensional integral cohomology class \(\mathcal{B}\) that differs from half the first Chern class of \(K\) by a torsion element, there are a nonempty set of closed integral curves of the Reeb vector field and a positive integer weight assigned to each curve in this set such that the resulting formal weighted sum of loops represents the Poincaré dual of \(\mathcal{B}\). To prove the theorem he uses ideas identifying the Seiberg-Witten and Gromov invariants of a compact, symplectic 4-manifold, and estimating the spectral flow of a family of Dirac operators on the 3-manifold \(\mathcal{M}\) as follows : Deform the Seiberg-Witten equations on the 3-manifold by adding a constant multiple \(r\) of \(-id\alpha\) to the curvature equation. Consider a sequence of values of \(r\) that tend to \(\infty\) and a corresponding sequence of solutions to the resulting equations. Under optimal circumstances, the spinor component of a solution to a large \(r\) of the equations is nearly zero on a set that closely approximates a closed integral curve of the Reeb vector field. Here the Seiberg-Witten Floer homology described by P. Kronheimer and T. Mrowka is used to prove that all larger \(r\) versions of the equations have solutions. As \(r\) goes to \(\infty\) along the sequence, the author shows that there is a subsequence of such sets that limits to the desired closed integral curve.


57R17 Symplectic and contact topology in high or arbitrary dimension
57R57 Applications of global analysis to structures on manifolds
57R58 Floer homology
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