## 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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### References:

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