A short history of the Weinstein conjecture
What determines whether periodic orbits exist for Hamiltonian flows on a fixed energy hypersurface? Alan Weinstein conjectured in 1978 that a geometric property of the hypersurface would provide a sufficient condition. Weinstein's conjecture, in successively refined forms, has provoked considerable attention to symplectic geometry, and led to fruitful interactions between analysis, topology and geometry. The conjecture has been proven in a number of special cases, but the general remains open. This paper provides a short expository overview of its history. The first important global results were proved by Rabinowitz (for star-shaped level sets) and Weinstein (for convex level sets): Every star-shaped (or convex) level set of a Hamiltonian function \(H\) in the standard symplectic manifold \(\mathbb{R}^{2n}\) admits a periodic orbity of \(X_H\). Weinstein then introduced the idea of a ``hypersurface of contact type'' -- generalizing star-shaped -- having the property of invariance under symplectic diffeomorphisms. A compact surface \(S\) in a symplectic manifold \((M,\omega)\) is said to be of contact type if there is a Liouville vector field \(Y\), defined in a neighborhood of \(S\), that is everywhere transverse to the hypersurface. Thus a more refined form of the conjecture emerged: Every compact hypersurface of a contact type in a symplectic manifold admits at least one closed characteristic. In hypersurfaces of contact type, one can always find a one-parameter family of such hypersurfaces that share the same dynamics. Viterbo then proved the conjecture for compact hypersurfaces on contact type in standard symplectic \(\mathbb{R}^{2n}\). A more general form of the conjecture -- an intrinsic version -- says that for every closed odd-dimensional manifold \(N\) with contact form \(\lambda\), the Reeb vector field \(X_\lambda\) admits a closed orbit. \textit{H. Hofer} proved the generalized conjecture for \(S^3\) in [Invent. Math. 114, No. 3, 515--563 (1993; Zbl 0797.58023)] and \textit{C. H. Taubes} proved it for any closed three-manifold in [Geom. Topol. 11, 2117--2202 (2007; Zbl 1135.57015)]. Since then, there have been several other proofs of special cases. The non-compact case remains challenging, with only a few limited results.
scientific article; zbMATH DE number 6112174