A note on Weinstein’s conjecture

Banyaga, Augustin

doi:10.1090/S0002-9939-1990-1021206-0

A note on Weinstein’s conjecture
HTML articles powered by AMS MathViewer

by Augustin Banyaga PDF

Proc. Amer. Math. Soc. 109 (1990), 855-858 Request permission

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.

References

David E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, Vol. 509, Springer-Verlag, Berlin-New York, 1976. MR 0467588, DOI 10.1007/BFb0079307
V. L. Ginzburg, New generalizations of Poincaré’s geometric theorem, Funktsional. Anal. i Prilozhen. 21 (1987), no. 2, 16–22, 96 (Russian). MR 902290
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, DOI 10.1007/978-1-4684-8670-4

Réduction symplectique et feuilletages Riemanniens: moment structural et théorème de convexité

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
Paul H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), no. 2, 157–184. MR 467823, DOI 10.1002/cpa.3160310203
Claude Viterbo, A proof of Weinstein’s conjecture in $\textbf {R}^{2n}$, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), no. 4, 337–356 (English, with French summary). MR 917741, DOI 10.1016/S0294-1449(16)30363-8
Paul H. Rabinowitz, Periodic solutions of a Hamiltonian system on a prescribed energy surface, J. Differential Equations 33 (1979), no. 3, 336–352. MR 543703, DOI 10.1016/0022-0396(79)90069-X