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 $\dot{x}=a\xi +bv,\dot{a}=0,a⪈0$. Let $L^{+}$, respectively $L^{-}$, be the set of curves in $C_{\beta }$ such that $b\ge 0$, respectively $b\le 0$. 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 ${\pi }_{n+1}(S^n)={\mathbb{Z}}_2,n\ge 3$. 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.