Taubes’s proof of the Weinstein conjecture in dimension three. (English) Zbl 1197.57023

Hutchings, Michael

Bull. Am. Math. Soc., New Ser. 47, No. 1, 73-125 (2010).

The Weinstein Conjecture asserts that on a closed oriented odd-dimensional manifold with a contact form, the associated Reeb vector field has a closed orbit. The author states that “the eventual goal of this article is to explain Taubes’s proof of the Weinstein conjecture in dimension three”. This he does well in the 12 sections of the paper.
In the first nine sections he discusses the topics and gives many examples relevant to the conjecture, introducing the reader to Hamiltonian vector fields on symplectic manifolds, contact forms and Reeb orbits, Morse functions and Morse homology, Spectral flow, Taubes’s Theorem on the equivalence of the Seiberg-Witten and Gromov invariants on \(4\)-manifolds, Seiberg-Witten Floer theory on \(3\)-manifolds, the Seiberg-Witten equations and gauge transformations.
In section 10 he gives an outline of Taubes’s proof and he fills in some details in section 11. Finally in section 12 he discusses possible generalizations, for example he points out that the only known examples of closed oriented (connected) \(3\)-manifolds with contact forms having only finitely many Reeb orbits are lens spaces and he asks whether every contact form on a closed oriented \(3\)-manifold other than a lens space or \(S^3\) has infinitely many embedded Reeb orbits.

Reviewer: Wolfgang Heil (Tallahassee)

MSC:

57R17	Symplectic and contact topology in high or arbitrary dimension
53D40	Symplectic aspects of Floer homology and cohomology
53D42	Symplectic field theory; contact homology
57-02	Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57N10	Topology of general \(3\)-manifolds (MSC2010)
57M50	General geometric structures on low-dimensional manifolds

Keywords:

Hamiltonian vector fields; symplectic manifolds; contact forms; Reeb orbits; Morse theory; spectral flow; Seiberg-Witten invariants; Gromov invariants; Floer theory; gauge theory

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