Viterbo, Claude A proof of Weinstein’s conjecture in \(\mathbb R^{2n}\). (English) Zbl 0631.58013 Ann. Inst. Henri Poincaré, Anal. Non Linéaire 4, 337-356 (1987). 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 Cited in 6 ReviewsCited in 55 Documents 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 Keywords:symplectic manifold; characteristic; existence of closed characteristics PDFBibTeX XMLCite \textit{C. Viterbo}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 4, 337--356 (1987; Zbl 0631.58013) Full Text: DOI Numdam EuDML References: [1] Bahri, A., Un problème variationnel sans compacité dans la géométrie de contact, C.R. Acad. Sc., T. 299, Série I, 757-760 (1984) · Zbl 0565.58018 [3] Berestycki, H.; Lasry, J. M.; Mancini, G.; Ruf, B., Existence of Multiple Periodic Orbits on Starshaped Hamiltonian Surfaces, Comm. Pure and Appl. Math., Vol. 38, 253-289 (1985) · Zbl 0569.58027 [4] Borel, A., Seminar on Transformation Groups, Annals of Math. Studies, No. 46 (1960), Princeton University Press: Princeton University Press New York [5] Clarke, F.; Ekeland, I., Hamiltonian Trajectories Having Prescribed Minimal Period, Comm. Pure and Appl. Math., Vol. 33, 103-113 (1980) · Zbl 0403.70016 [7] Fadell, E. R.; Rabinowitz, P. H., Generalized Cohomological Index Theories for Lie Group Action with an Application to Bifurcation Questions for Hamiltonian Systems, Invent. Math., Vol. 45, 139-174 (1978) · Zbl 0403.57001 [9] Rabinowitz, P. H., Periodic Solutions of Hamiltonian Systems, Comm. Pure and Appl. Math., Vol. 31, 157-184 (1978) · Zbl 0358.70014 [10] Seifert, H., Periodische Bewegungen mechanischer Systeme, Math. Z., Vol. 51, 197-216 (1948) · Zbl 0030.22103 [11] Weinstein, A., Periodic Orbits for Convex Hamiltonian Systems, Ann. of Math., Vol. 108, 507-518 (1978) · Zbl 0403.58001 [12] Weinstein, A., On the hypotheses of Rabinowitz’ periodic orbit theorem, Journal of Diff. Eq., Vol. 33, 353-358 (1979) · Zbl 0388.58020 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.