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A proof of Weinstein’s conjecture in \(\mathbb R^{2n}\). (English) Zbl 0631.58013

Let \((M^{2n},\omega)\) be a symplectic manifold, \(\Sigma\) a hypersurface of \(M^{2n}\). \(\Sigma\) is said to be of contact type if there is a 1- form \(\theta\) on \(\Sigma\), such that \(d\theta =j^*\omega\) (j: \(\Sigma\to W\) is the inclusion map), and \(\theta \wedge (d\theta)^{n- 1}\) is a volume form on \(\Sigma\). A characteristic is a curve everywhere tangent to the line field ker \(\omega|_{\Sigma}\). Theorem. If \(\Sigma \subset ({\mathbb{R}}^{2n},\omega_ 0)\) is a compact hypersurface of contact type, then \(\Sigma\) has a closed characteristic (here \(\omega_ 0=\sum^{n}_{i=1}dx_ i\wedge dy^ i).\)
The starting point of the investigation was Seifert’s result (1948) on the existence of closed characteristics for some special class of convex hypersurfaces. Weinstein extended this result to general \(C^ 2\) convex hypersurfaces and made the following conjecture: if \(\Sigma\subset (M,\omega)\) is a compact hypersurface of contact type and \(H^ 1(\Sigma,{\mathbb{R}})=0\), then \(\Sigma\) has a closed characteristic.
Reviewer: I.Ya.Dorfman

MSC:

53D05 Symplectic manifolds (general theory)
53D10 Contact manifolds (general theory)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C40 Global submanifolds
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