Related papers: Exponential Convergence to Non-degenerate Reeb Cho…

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…

We prove that every closed, connected contact 3-manifold can be obtained from the 3-sphere with its standard contact structure by contact surgery of coefficient plus or minus 1 along a Legendrian link. As a corollary, we derive a result of…

Consider a symplectic surface in a three-dimensional contact manifold with boundary on Reeb orbits (periodic orbits of the Reeb vector field). We assume that the rotation numbers of the boundary Reeb orbits satisfy a certain inequality, and…

In this note we prove an effective characterization of when two finite-degree covers of a connected, orientable surface of negative Euler characteristic are isomorphic in terms of which curves have simple elevations, weakening the…

The symplectic cohomology of certain symplectic manifolds $W$ with non-compact ends modelled on the positive symplectization of a compact contact manifold $Y$ is shown to vanish whenever there is a positive loop of contactomorphisms of $Y$…

This paper and its sequel prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves u_+ and u_- in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit gamma, the total…

We classify convex disks with a fixed characteristic foliation and Legendrian boundary, up to contact isotopy relative to the boundary, in every closed overtwisted contact 3-manifold. This classification covers cases where the neighborhood…

In this article, we first give a proof on the Arnold chord conjecture which states that every Reeb flow has at least as many Reeb chords as a smooth function on the Legendre submanifold has critical points on contact manifold. Second, we…

Let $(M, \alpha)$ be a $2n+1$-dimensional connected compact contact toric manifold of Reeb type. Suppose the contact form $\alpha$ is regular, we find conditions under which $M$ is homeomorphic to $S^{2n+1}$.

This note explores the use of Newton polytopes in the study of Lagrangian fillings of Legendrian submanifolds. In particular, we show that Newton polytopes associated to augmented values of Reeb chords can distinguish infinitely many…

This paper is the sequel to the previous paper [Ne15], which showed that sufficient regularity exists to define cylindrical contact homology in dimension three for nondegenerate dynamically separated contact forms, a subclass of dynamically…

We present a new example of a finite-dimensional noncommutative manifold, namely the noncommutative cylinder. It is obtained by isospectral deformation of the canonical triple associated to the Euclidean cylinder. We discuss Connes'…

We extend classical results by Lavrent'ev and Kufarev concerning the product of the conformal radii of planar non-overlapping domains. We also extend relatively recent results for the case of domains in the $n$-dimensional Euclidean space,…

We study the asymptotic behaviour of a real-valued diffusion whose non-regular drift is given as a sum of a dissipative term and a bounded measurable one. We prove that two trajectories of that diffusion converge a.s. to one another at an…

Let $\mathbb{R} \times (P \times \mathbb{R})$ be the symplectisation of the contactisation of an exact symplectic manifold $P$, and let $\mathbb{R} \times \Lambda$ be a cylinder over a Legendrian submanifold in the contactisation. We show…

We study holomorphic spheres in certain symplectic cobordisms and derive information about periodic Reeb orbits in the concave end of these cobordisms from the non-compactness of the relevant moduli spaces. We use this to confirm the strong…

Let $\Sigma$ be a surface with a symplectic form, let $\phi$ be a symplectomorphism of $\Sigma$, and let $Y$ be the mapping torus of $\phi$. We show that the dimensions of moduli spaces of embedded pseudoholomorphic curves in $\R\times Y$,…

In this paper we study holomorphic Legendrian curves in the standard holomorphic contact structure on $\mathbb{C}^{2n+1}$ for any $n\in\mathbb{N}$. We provide several approximation and desingularization results which enable us to prove…

Let $\Lambda^{\pm} = \Lambda^{+} \cup \Lambda^{-} \subset (\mathbb{R}^{3}, \xi_{std})$ be a contact surgery diagram determining a closed, connected contact $3$-manifold $(S^{3}_{\Lambda^{\pm}}, \xi_{\Lambda^{\pm}})$ and an open contact…

We find convergent double series expansions for Legendre's third incomplete elliptic integral valid in overlapping subdomains of the unit square. Truncated expansions provide asymptotic approximations in the neighbourhood of the logarithmic…