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.
Highlights
This paper is concerned with two issues; it is, sub-divided into two distinct parts
The result that we prove in this paper reads as follows: Theorem 1.1 Considering a deformation of contact forms αt under (A)t, there is a corresponding deformation of decreasing “pseudo-gradients” Zt for the variational problems (Jt, Zt )—we assume both to be in general position—so that the following hold: (i) Let ∂per be the intersection operator restricted to periodic orbits
2 Part I: The ∂per and ∂∞ operators. We prove on this first part Theorems 1.1 and 1.2 stated above and we compute the value of the homology of ∂per starting from the standard contact form and the first exotic contact form of Gonzalo and Varela [14] on S3
Summary
This paper is concerned with two issues; it is, sub-divided into two distinct parts. The result that we prove in this paper reads as follows: Theorem 1.1 Considering a deformation of contact forms αt under (A)t , there is a corresponding deformation of decreasing “pseudo-gradients” Zt for the variational problems (Jt , Zt )—we assume both to be in general position—so that the following hold: (i) Let ∂per be the intersection operator restricted to periodic orbits This operator is not modified by tangencies and creations or cancellations involving the critical points at infinity of (J, Cβ ). We conjecture that the result ∂per ◦∂per = 0 holds in almost full generality at the odd indexes to the least The proof of this conjecture would involve in a first step the study, at the time zero of the deformation, that is e.g. for the first exotic contact form of Gonzalo and Varela [14], of the Morse relation ∂ y2(∞k ) = c2k−1 + x2∞k−,t1(∗) described above. This section allows to understand that, for “triangles” of dominations x2k+1/x2∞k /x2k−1 to arise, x2∞k must be built with a combination of critical points at infinity—a single one will not “work”—and the Morse relation between x2k+1 and x2∞k must be of “point to circle” type
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