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Zbl 0631.58013
Viterbo, Claude
A proof of Weinstein's conjecture in $\Bbb R^{2n}$. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 4, 337-356 (1987). ISSN 0294-1449

Let $(M\sp{2n},\omega)$ be a symplectic manifold, $\Sigma$ a hypersurface of $M\sp{2n}$. $\Sigma$ is said to be of contact type if there is a 1- form $\theta$ on $\Sigma$, such that $d\theta =j\sp*\omega$ (j: $\Sigma\to W$ is the inclusion map), and $\theta \wedge (d\theta)\sp{n- 1}$ is a volume form on $\Sigma$. A characteristic is a curve everywhere tangent to the line field ker $\omega\vert\sb{\Sigma}$. Theorem. If $\Sigma \subset ({\bbfR}\sp{2n},\omega\sb 0)$ is a compact hypersurface of contact type, then $\Sigma$ has a closed characteristic (here $\omega\sb 0=\sum\sp{n}\sb{i=1}dx\sb i\wedge dy\sp i).$ \par 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 $C\sp 2$ convex hypersurfaces and made the following conjecture: if $\Sigma\subset (M,\omega)$ is a compact hypersurface of contact type and $H\sp 1(\Sigma,{\bbfR})=0$, then $\Sigma$ has a closed characteristic.
[I.Ya.Dorfman]

MSC 2000:: *53D05 Symplectic manifolds, general
53D10 Contact manifolds, general
53C15 Geometric structures on manifolds
53C40 Submanifolds (differential geometry)

Keywords: symplectic manifold; characteristic; existence of closed characteristics

Cited in: Zbl 1066.53138 Zbl 1121.53061 Zbl 0978.53135 Zbl 0894.53035 Zbl 0668.58044

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