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A note on Weinstein's conjecture

Author: Augustin Banyaga
Journal: Proc. Amer. Math. Soc. 109 (1990), 855-858
MSC: Primary 58F22; Secondary 58F05, 58F18
MathSciNet review: 1021206
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Abstract: We prove that the contact foliation of a compact contact manifold $ \left( {M,\alpha } \right)$ has at least one compact leaf in the following two cases: (i) $ \alpha $ is a $ K$-contact form and $ M$ is simply connected, (ii) $ \alpha $ is $ {C^2}$-close to a regular contact form. This solves the Weinstein conjecture in those particular cases.

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  • [1] David E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, Vol. 509, Springer-Verlag, Berlin-New York, 1976. MR 0467588
  • [2] V. L. Ginzburg, New generalizations of Poincaré’s geometric theorem, Funktsional. Anal. i Prilozhen. 21 (1987), no. 2, 16–22, 96 (Russian). MR 902290
  • [3] Pierre Molino, Riemannian foliations, Progress in Mathematics, vol. 73, Birkhäuser Boston, Inc., Boston, MA, 1988. Translated from the French by Grant Cairns; With appendices by Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu. MR 932463
  • [4] -, Réduction symplectique et feuilletages Riemanniens: moment structural et théorème de convexité, in Séminaire Gaston Darboux de Géometrie et Topologie Differentielle, 1987-88, Univ. Montpellier, France, pp. 11-25.
  • [5] Gilbert Monna, Feuilletages de 𝐾-contact sur les variétés compactes de dimension 3, Publ. Sec. Mat. Univ. Autònoma Barcelona 28 (1984), no. 2-3, 81–87 (French). MR 857078
  • [6] Paul H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), no. 2, 157–184. MR 0467823,
  • [7] Claude Viterbo, A proof of Weinstein’s conjecture in 𝑅²ⁿ, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), no. 4, 337–356 (English, with French summary). MR 917741
  • [8] Alan Weinstein, On the hypotheses of Rabinowitz’ periodic orbit theorems, J. Differential Equations 33 (1979), no. 3, 353–358. MR 543704,

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Keywords: $ K$-contact form, contact foliation, Riemannian foliation, transverse symplectic structure, characteristics
Article copyright: © Copyright 1990 American Mathematical Society

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