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Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three. (English) Zbl 0797.58023

A contact form on the smooth closed orientable \((2n + 1)\)-dimensional manifold \(M\) is a 1-form \(\lambda\) such that \(\lambda \wedge (d \lambda)^ n\) is a volume form on \(M\). Associated to \(\lambda\) there are two important structures: the Reeb vector field \(X = X_ \lambda\) defined by \(i_ X \lambda \equiv 1\), \(i_ X d\lambda \equiv 0\) and the contact structure \(\xi = \xi_ \lambda \to M\) given by \(\xi_ \lambda = \text{Kern}(\lambda) \subset TM\). The contact structure is stable [see J. W. Gray, Ann. Math., II. Ser. 69, 421-450 (1959; Zbl 0092.39301)], but in contrast the dynamics of the Reeb vector field changes drastically under small perturbation. It is known the Weinstein conjecture [A. Weinstein, J. Differ. Equations 33, 353-358 (1979; Zbl 0388.58020)]: for every contact form on a closed manifold \(M\) the Reeb vector field has at least one periodic orbit, provided \(H^ 1(M;R) = 0\). The Weinstein conjecture holds for \(M\) provided for every contact form \(\lambda\) on \(M\) the associated Reeb vector field has a closed orbit. The author is restricted to the case \(\dim M = 3\), since in three dimensions his method seems to be most powerful and gives a more complete picture of the Weinstein conjecture. One of the main results is the existence of periodic orbits for the Reeb vector field associated to overtwisted contact forms. For such contact forms there does not exist any compact symplectic manifold \((W,\omega)\) with \(M = \partial W\) and \(d \lambda = \omega\mid M\). There are for example many overtwisted 1-forms on \(S^ 3\). As a corollary \(S^ 3\) with such structure can never be embedded into \(\mathbb{R}^ 4\) in such a way that the Hamiltonian flow on the embedded \(S^ 3\) is the conjugate to the abstract Reeb flow. The problem for higher dimensions and some applications will be given in a future paper.

MSC:

37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
53D35 Global theory of symplectic and contact manifolds
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
53D10 Contact manifolds (general theory)
32Q65 Pseudoholomorphic curves
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References:

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