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"

TL;DR

This paper proves the Weinstein Conjecture on S^3 by establishing an equivariant variational framework involving linking cycles in the space of Legendrian curves, revealing the existence of periodic Reeb orbits.

Contribution

It introduces a novel linking/equivariant homology approach using Morse theory and degree theory to prove the existence of periodic orbits in contact geometry, extending Rabinowitz's linking argument.

Findings

Infinite cycles in equivariant homology imply periodic orbits

Unions of unstable manifolds at infinity include periodic orbits

The method applies to overtwisted contact forms on S^3

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.