Related papers: Contact forms with large systolic ratio in arbitra…
By a result of Eliashberg, every symplectic filling of a three-dimensional contact connected sum is obtained by performing a boundary connected sum on another symplectic filling. We prove a partial generalization of this result for…
Gromov's famous non-squeezing theorem (1985) states that the standard symplectic ball cannot be symplectically squeezed into any cylinder of smaller radius. Does there exist an analogue of this result in contact geometry? Our main finding…
We present new explicit tight and overtwisted contact structures on the (round) 3-sphere and the (flat) 3-torus for which the ambient metric is weakly compatible. Our proofs are based on the construction of nonvanishing curl eigenfields…
The systole of a compact non simply connected Riemannian manifold is the smallest length of a non-contractible closed curve ; the systolic ratio is the quotient $(\mathrm{systole})^n/\mathrm{volume}$. Its supremum on the set of all the…
We study codimension $1$ embeddings preserving open book structures. In particular, we prove that every closed orientable 3-manifold admits a codimension-1 spun embedding in a finite connected sum of $S^2 \times S^2$s and $S^2…
We discuss embedding of manifolds in the category of open books, contact manifolds and contact open books. We prove an open book version of the Haefliger--Hirsch embedding theorem by showing that every $k$-connected closed $n$-manifold…
An oriented compact closed manifold is called inflexible if the set of mapping degrees ranging over all continuous self-maps is finite. Inflexible manifolds have become of importance in the theory of functorial semi-norms on homology.…
We use Lerman's contact cut construction to find a sufficient condition for Hamiltonian diffeomorphisms of compact surfaces to embed into a closed 3-manifold as Poincar\'e return maps on a global surface of section for a Reeb flow. In…
We extend a result of M. Katz on conformal systoles to all four-manifolds with b^+=1 which have odd intersection form. The same result holds for all four-manifolds with b^+=1 with even intersection form and which are symplectic or satisfy…
We show that the existence of one simple closed Reeb orbit of a particular type (a symplectically degenerate maximum) forces the Reeb flow to have infinitely many periodic orbits. We use this result to give a different proof of a recent…
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 show that the length $R$ of a systole of a closed hyperbolic $n$-manifold $(n \geq 3)$ admitting a triangulation by $t$ $n$-simplices can be bounded below by a function of $n$ and $t$, namely \[ R \geq \frac{1}{2^{(nt)^{O(n^4t)} }} .\]…
In "Proof of the Arnold chord conjecture in three dimensions I", we deduced the Arnold chord conjecture in three dimensions from another result, which asserts that an exact symplectic cobordism between contact three-manifolds induces a map…
A contact manifold is a manifold equipped with a distribution of codimension one that satisfies a `maximal non-integrability' condition. A standard example of a contact structure is a strictly pseudoconvex CR manifold, and operators of…
We construct contact forms with constant $Q^\prime$-curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by…
A real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution, called a real structure. In this article we study open book decompositions on smooth real 3-manifolds that are compatible with the real…
Starting from a cubic form, we give a general construction of a quasi-complete homogeneous manifold endowed with a natural contact structure. We show that it can be compactified into a projective contact manifold if and only if the cubic…
We show that contact homology distinguishes infinitely many tight contact structures on any orientable, toroidal, irreducible 3-manifold. As a consequence of the contact homology computations, on a very large class of toroidal manifolds,…
On every compact, orientable, irreducible 3-manifold V which is toroidal or has torus boundary components we construct a contact 1-form whose Reeb vector field R does not have any contractible periodic orbits and is tangent to the boundary.…
We show that any codimension one hyperbolic attractor of a diffeomorphism of a (d+1)-dimensional closed manifold is shape equivalent to a (d+1)-dimensional torus with a finite number of points removed, or, in the non-orientable case, to a…