Related papers: A counterexample to the singular Weinstein conject…

The original Arnold chord conjecture states that every closed Legendrian submanifold of the standard contact sphere $S^{2n-1}$ admits a Reeb chord with distinct endpoints with respect to any contact form. In this paper, we prove this…

Let M be a smooth closed manifold and T*M its cotangent bundle endowed with the usual symplectic structure. A hypersurface S in T*M is said to be fiberwise starshaped if for each point q in M the intersection of S with the fiber at q is…

We prove the strong Weinstein conjecture for closed contact manifolds that appear as the concave boundary of a symplectic cobordism admitting an essential local foliation by holomorphic spheres.

We study the contact geometry of the connected components of the energy hypersurface, in the symmetric restricted 3-body problem on $\mathbb{S}^2$, for a specific type of motion of the primaries. In particular, we show that these components…

Let $(Y,\lambda)$ be a non-degenerate contact three manifold. D. Cristfaro-Gardiner, M. Hutshings and D. Pomerleano showed that if $c_{1}(\xi=\mathrm{Ker}\lambda)$ is torsion, then the Reeb vector field of $(Y,\lambda)$ has infinity many…

We exhibit transverse knot types on the standard contact $3$-sphere that cannot be realized as periodic Reeb orbits of a dynamically convex contact form. In particular, such transverse knot types do not arise as closed characteristics of…

We prove the existence of periodic orbits for steady $C^\omega$ Euler flows on all Riemannian solid tori. By using the correspondence theorem from part I of this series, we reduce the problem to the Weinstein Conjecture for solid tori. We…

It is well known that the Eisenbud-Goto regularity conjecture is true for arithmetically Cohen-Macaulay varieties, projective curves, smooth surfaces, smooth threefolds in $\mathbb{P}^5$, and toric varieties of codimension two. After J.…

We prove that a Weinstein domain symplectically embedded in a closed symplectic manifold always admits symplectic hypersurfaces in its complement, possibly after a deformation. As a consequence, we obtain an obstruction for a closed…

We survey some results on the existence (and non-existence) of periodic Reeb orbits on contact manifolds, both in the open and closed case. We place these statements in the context of Finsler geometry by including a proof of the folklore…

In this paper, it is proved that under dynamically convex condition, there exist at least $[\frac{n+1}{2}]$ closed Reeb orbits on a closed contact type hypersurface in $T^*S^n$ enclosing the zero section and bounding a simply connected…

We prove that every Reeb flow on a closed connected three-manifold has either two or infinitely many simple periodic orbits, assuming that the associated contact structure has torsion first Chern class. As a special case, we prove a…

An estimate on the number of distinct relative periodic orbits around a stable relative equilibrium in a Hamiltonian system with continuous symmetry is given. This result constitutes a generalization to the Hamiltonian symmetric framework…

We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. In particular, let $\phi$ be a rational function with no superattracting periodic points other than exceptional points. If the coefficients of $\phi$…

In this paper, we prove that every graphical hypersurface in a prequantization bundle over a symplectic manifold $M$, pinched between two circle bundles whose ratio of radii is less than $\sqrt{2}$ carries either one short simple periodic…

The first two authors showed in~\cite{AM1} how the Conley-Zehnder index of any contractible periodic Reeb orbit of a non-degenerate toric contact form on a good toric contact manifold with zero first Chern class, i.e. a Gorenstein toric…

We show that every quasiconformal contact foliation supports an invariant metric and characterise such foliations by the dynamical property of $C^1$-equicontinuity. We prove that a generalisation of the Weinstein conjecture holds for…

The monodromy conjecture is an umbrella term for several conjectured relationships between poles of zeta functions, monodromy eigenvalues and roots of Bernstein-Sato polynomials in arithmetic geometry and singularity theory. Even the…

We investigate a question of Cooper adjacent to the Virtual Haken Conjecture. Assuming certain conjectures in number theory, we show that there exist hyperbolic rational homology 3-spheres with arbitrarily large injectivity radius. These…

We prove that every nondegenerate contact form on a closed connected three-manifold, such that the associated contact structure has torsion first Chern class, has either two or infinitely many simple Reeb orbits. By previous results it…