A Linking/$S^1$ Equivariant Variational Argument in the space of Dual Legendrian curves and the proof of the Weinstein Conjecture on $S^3$ "in the large"

Abstract

Let $\alpha$ be a contact form on $S^3$, let $\xi$ be its Reeb vector-field and let $v$ be a non-singular vector-field in $ker\alpha$. Let $C_\beta$ be the space of curves $x$ on $S^3$ such $\dot x=a\xi+bv, \dot a=0, a \gneq 0$. Let $L^+$, respectively $L^-$, be the set of curves in $C_\beta$ such that $b\geq 0$, respectively $b \leq 0$. Let, for $x \in C_\beta$, $J(x)=\int_0^1\alpha_x(\dot x)dt$. We establish in this paper that an infinite number of cycles in the $S^1$-equivariant homology of $C_\beta$,{\bf relative} to $L^+ \cup L^-$ and to some specially designed "bottom set", see section 4, are achieved in the Morse complex of $(J, C_\beta)$ by unions of unstable manifolds of critical points (at infinity)which must include periodic orbits of $\xi$; ie unions of unstable manifolds of critical points at infinity alone cannot achieve these cycles. The topological argument of existence of a periodic orbit for $\xi$ turns out to be surprisingly close, in spirit, to the linking/equivariant argument of P.H. Rabinowitz in [12]. The objects and the frameworks are strikingly different, but the original proof of [12] can be recognized in our proof, which uses degree theory, the Fadell-Rabinowitz index [8] and the fact that $\pi_{n+1}(S^n)=\mathbb{Z}_2, n\geq 3$. The arguments hold under the basic assumption that no periodic orbit of index $1$ connects $L^+$ and $L^-$. To a certain extent, the present result runs, especially in the case of three-dimensional overtwisted [8] contact forms, against the existence of non-trivial algebraic invariants defined by the periodic orbits of $\xi$ and independent of what $ker \alpha$ and/or $\alpha$ are.