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Arnold conjectured that every Legendrian knot in the standard contact structure on the 3-sphere possesses a haracteristic chord with respect to any contact form. I confirm this conjecture if the know has Thurston-Bennequin invariant $-1$. More generally, existence of chords is proved for a standard Legendrian unknot on the boundary of a subcritical Stein manifold of any dimension. There is also a multiplicity result which implies in some situations existence of infinitely many chords. The proof relies...

In this paper we study topological properties of stable Hamiltonian structures. In particular, we prove the following results in dimension three: The space of stable Hamiltonian structures modulo homotopy is discrete; stable Hamiltonian structures are generically Morse-Bott (i.e. all closed orbits are Bott nondegenerate) but not Morse; the standard contact structure on $S^3$ is homotopic to a stable Hamiltonian structure which cannot be embedded in ${ℝ}^4$. Moreover, we derive a structure theorem for stable...

The first two authors have recently defined Rabinowitz Floer homology groups $RF{H}_{*}(M,W)$ associated to a separating exact embedding of a contact manifold $(M,\xi )$ into a symplectic manifold $(W,\omega )$. These depend only on the bounded component $V$ of $W\setminus M$. We construct a long exact sequence in which symplectic cohomology of $V$ maps to symplectic homology of $V$, which in turn maps to Rabinowitz Floer homology $RF{H}_{*}(M,W)$, which then maps to symplectic cohomology of $V$. We compute $RF{H}_{*}(S{T}^{*}L,T^{*}L)$, where $S{T}^{*}L$ is the unit cosphere bundle of a closed manifold...

We give an example of a symplectic manifold with a stable hypersurface such that nearby hypersurfaces are typically unstable.