The Weinstein conjecture on \(C^1\)-smooth hypersurface of contact type. (Q1856783)

Weinstein's conjecture [\textit{A. Weinstein}, J. Differ. Equations 33, 353--358 (1979; Zbl 0388.58020)] claims that every smooth hypersurface of contact type carries a closed characteristic in symplectic manifolds, where a smooth compact orientable hypersurface \(S\subset (M,\omega)\) is said to be of contact type if there exists a smooth vector field \(X\) defined on a neighborhood \(U\) of \(S\) which is transversal to \(S\) and satisfies \(L_X\omega=\omega\) on \(U\) and ``a characteristic'' is a leaf of the characteristic foliation. The conjecture holds in many symplectic manifolds, for example in \(\mathbb R^{2n}\) was proved by \textit{C. Viterbo} [Ann. Inst. H. Poincaré, Anal. Non Linéaire 4, 337--356 (1987; Zbl 0631.58013)]. The study of Weinstein's conjecture led to the introduction of an important symplectic invariant, Hofer-Zehnder symplectic capacity \(c_{\text{HZ}}\). In [Symplectic invariants and Hamiltonian dynamics (Birkhäuser: Basel) (1994; Zbl 0805.58003)] \textit{H. Hofer} and \textit{E. Zehnder} proved that if a smooth hypersurface of contact type on a symplectic manifold has an open neighborhood with finite capacity \(c_{\text{HZ}}\) then it carries at least a closed characteristic. The author of this paper asks if the Weinstein conjecture also holds for \(C^1\)-smooth hypersurfaces of contact type. A symplectic capacity of Hofer-Zehnder type \(c_{\text{HZ}}^1\) that is invariant under \(C^1\)-symplectomorphisms satisfying similar properties to \(c_{\text{HZ}}\) is introduced and and used to prove some results on \(C^1\)-Weinstein conjecture. As a consequence some symplectic results are generalized in the \(C^1\) context, as for example the Gromov's nonsqueezing theorem.