Related papers: A counterexample to the singular Weinstein conject…

This paper and its sequel prove that every Legendrian knot in a closed three-manifold with a contact form has a Reeb chord. The present paper deduces this result from another theorem, asserting that an exact symplectic cobordism between…

We generalize the Weinstein-Moser theorem on the existence of nonlinear normal modes (i.e., periodic orbits) near an equilibrium in a Hamiltonian system to a theorem on the existence of relative periodic orbits near a relative equilibrium…

In a neighborhood of a hyperbolic periodic orbit of a volume-preserving flow on a manifold of dimension 3, we define and show the existence of a normal form for the generator of the flow that encodes the dynamics. If the flow is a contact…

We show that any co-oriented closed contact manifold of dimension at least five admits a contact form such that the contact volume is arbitrarily small but the Reeb flow admits a global hypersurface of section with the property that the…

In this paper we prove the existence of real-analytic natural Hamiltonian systems - i.e. where H(q,p)=T(q,p)+V(q) in the 2N-dimensional real space, where N is any integer greater than 1 - with non critical energy levels E for the potential…

For Reeb vector fields on closed 3-manifolds, cylindrical contact homology is used to show that the existence of a set of closed Reeb orbit with certain knotting/linking properties implies the existence of other Reeb orbits with other…

We prove the Conley conjecture for negative monotone, closed symplectic manifolds, i.e., the existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms of such manifolds.

The Reeb space of a generic map is the space of all connected components of preimages of the map. Reeb spaces are fundamental and useful tools in the theory of Morse functions and higher dimensional variants and their applications to…

We characterise boundary shaped disc like neighbourhoods of certain isotropic submanifolds in terms of aperiodicity of Reeb flows. We prove uniqueness of homotopy and diffeomorphism type of such contact manifolds assuming non-existence of…

We prove that for a certain class of closed monotone symplectic manifolds any Hamiltonian diffeomorphism with a hyperbolic fixed point must necessarily have infinitely many periodic orbits. Among the manifolds in this class are complex…

A Reeb flow on a contact manifold is called Besse if all its orbits are periodic, possibly with different periods. We characterize contact manifolds whose Reeb flows are Besse as principal S^1-orbibundles over integral symplectic orbifolds…

Through the use of sub-Riemannian metrics we provide quantitative estimates for the maximal tight neighbourhood of a Reeb orbit on a three-dimensional contact manifold. Under appropriate geometric conditions we show how to construct closed…

A long-standing conjecture in Hamiltonian Dynamics states that the Reeb flow of any convex hypersurface in $\mathbb{R}^{2n}$ carries an elliptic closed orbit. Two important contributions toward its proof were given by Ekeland in 1986 and…

We survey various recent results on the existence and properties of periodic orbits of Reeb vector fields in three dimensions. We give an introduction to the "elementary spectral invariants" of contact three-manifolds, and we explain how…

In this paper, we provide new and simpler proofs of two theorems of Gluck and Harrison on contact structures induced by great circle or line fibrations. Furthermore, we prove that a geodesic vector field whose Jacobi tensor is parallel…

We prove that if a contact 3-manifold admits an open book decomposition of genus 0, a certain intersection pattern cannot appear in the homology of any of its minimal symplectic fillings, and moreover, fillings cannot contain symplectic…

When a real saddle equilibrium in a three-dimensional vector field undergoes a homoclinic bifurcation, the associated two-dimensional invariant manifold of the equilibrium closes on itself in an orientable or non-orientable way. We are…

We show that the presence of a non-contractible one-periodic orbit of a Hamiltonian diffeomorphism of a connected closed symplectic manifold $(M,\omega)$ implies the existence of infinitely many non-contractible simple periodic orbits,…

A contact form is called K-contact if its Reeb vector field is Killing with respect to some Riemannian metric. In this paper we classify K-contact forms whose Reeb vector field admits at least one non-periodic orbit, on three-dimensional…

We show that whenever a closed symplectic manifold admits a Hamiltonian diffeomorphism with finitely many simple periodic orbits, the manifold has a spherical homology class of degree two with positive symplectic area and positive integral…