Abstract
We address in this paper the Fredholm and compactness issues for the variational problem (J,C β ), Bahri (Pitman Research Notes in Mathematics Series No. 173. Scientific and Technical, London, 1988), Bahri (C. R. Acad. Sci. Paris 299, Serie I, 15, 757–760, 1984). We prove that the intersection operator restricted to periodic orbits of the Reeb vector-field ∂per does not mix with the intersection operator ∂∞ of the critical points at infinity. The Fredholm issues are extensively discussed in the Introduction and solved in Bahri (Arab J Maths, 2014). We also address in this paper the issue of existence of periodic orbits for three-dimensional Reeb vector-fields, the Weinstein conjecture (Weinstein, J Differ Equ 33:353–358, 1979) on S3, solved in dimension 3 throughout the works of Rabinowitz (Commun Pure Appl Math 31:157–184, 1978) and Hofer (Invent Math 114:515–563, 1993); see also Hutchings (Proc. 2010 ICM 46:1022–1041, 2010) and Taubes (Geom Topol 11:2117–2202, 2007) for the full Weinstein conjecture in dimension 3. Following our previous work (Bahri, Adv Nonlinear Stud 8:1–17, 2008), we devise a new method to find these periodic orbits when they are of odd index. We conjecture that this method, when combined with the other results described above about the intersection operator, gives rise to a homology that is specific of the contact structure and that is invariant by deformation. The existence result, as derived here, is weaker than the one announced by Taubes (Geom Topol 11:2117–2202, 2007). After appropriate generalization, it provides a new proof, via variational theory, of the Weinstein conjecture on three-dimensional closed contact manifolds with finite fundamental group.
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Per Diana Nunziante, un caro ricordo “Te voglio bene assaie”. Grazie, grazie tanto!
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Bahri, A. Morse relations and Fredholm deformations of v-convex contact forms. Arab. J. Math. 3, 93–187 (2014). https://doi.org/10.1007/s40065-014-0098-1
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DOI: https://doi.org/10.1007/s40065-014-0098-1