Related papers: Universal orderability of Legendrian isotopy class…

It is shown that if the universal cover of a manifold $M$ is an open manifold, then two different fibres of the spherical cotangent bundle $ST^*M$ cannot be connected by a non-negative Legendrian isotopy. This result is applied to the study…

Positive loops of Legendrian embeddings are examined from the point of view of Floer homology of Lagrangian cobordisms. This leads to new obstructions to the existence of a positive loop containing a given Legendrian, expressed in terms of…

We construct new unbounded invariant distances on the universal cover of certain Legendrian isotopy classes. This is the first instance where unboundedness of an invariant distance is obtained without assuming the existence of a positive…

Let $\{\Lambda^\infty_t\}$ be an isotopy of Legendrians (possibly singular) in a unit cosphere bundle $S^*M$. Let $Sh(M, \Lambda^\infty_t)$ be the differential graded (dg) derived category of constructible sheaves on $M$ with singular…

We study the notion of orderability of isotopy classes of Legendrian submanifolds and their universal covers, with some weaker results concerning spaces of contactomorphisms. Our main result is that orderability is equivalent to the…

The classical Bott-Samelson theorem states that if on a Riemannian manifold all geodesics issuing from a certain point return to this point, then the universal cover of the manifold has the cohomology ring of a compact rank one symmetric…

We prove a structure theorem for compact aspherical Lorentz manifolds with abundant local symmetry. If M is a compact, aspherical, real-analytic, complete Lorentz manifold such that the isometry group of the universal cover has semisimple…

We show that Legendrian pre-quantization lifts of many non-exact Lagrangian submanifolds in $\mathbb{C}^n$ retain some quantitative rigidity from the symplectic base. In particular, they cannot be moved by Legendrian isotopy into an…

We formulate conjectures generalizing some known results to the category of virtual Legendrian knots. This includes statements relating virtual Legendrian knots to ordinary Legendrian knots, non-existence of positive virtual Legendrian self…

Using convex surfaces and Kanda's classification theorem, we classify Legendrian isotopy classes of Legendrian linear curves in all tight contact structures on $T^3$. Some of the knot types considered in this article provide new examples of…

An elementary stabilization of a Legendrian link $L$ in the spherical cotangent bundle $ST^*M$ of a surface $M$ is a surgery that results in attaching a handle to $M$ along two discs away from the image in $M$ of the projection of the link…

In this paper, via h-principle we prove that there exist contractible positive loops of Legendrian embeddings based at any loose Legendrian submanifold. As an application, we define a new partial order on $\widetilde{Cont}_0(M,\xi)$ and…

We show that there is no positive loop inside the component of a fiber in the space of Legendrian embeddings in the contact manifold $ST^*M$, provided that the universal cover of $M$ is $\RM^n$. We consider some related results in the space…

Viterbo has conjectured that any Lagrangian in the unit co-disc bundle of a torus which is Hamiltonian isotopic to the zero-section satisfies a uniform bound on its spectral norm; a recent result by Shelukhin showed that this is indeed the…

In this paper Legendrian graphs in $(\mathbb{R}^3,\xi_{\mathrm{st}})$ are considered modulo Legendrian isotopy and edge contraction. To a Legendrian graph we associate a (generalized) rectangular diagram --- a purely combinatorial object.…

The main theorem characterizes all Legendrian negative torus knots in universally tight lens space in the sense of coarse equivalence. Together with Onaran's results on Legendrian positive torus knots, all Legendrian torus knots in…

Let $M$ be a closed manifold. We introduce a family of Legendrian isotopy invariants for Legendrians in $J^1M$, which we collectively call Legendrian higher torsion. Given a choice of a class $\mathcal{F}$ of fibre bundles over $M$,…

We prove, for a class of contact manifolds, that the universal cover of the group of contact diffeomorphisms carries a natural partial order. It leads to a new viewpoint on geometry and dynamics of contactomorphisms. It gives rise to…

We describe a semi-local canonical form for Legendrian foliations on contact manifolds in the neighbourhood of a Legendrian submanifold. This result generalizes local results by Libermann and Pang on Legendrian foliations on contact…

An exact Lagrangian submanifold $L$ in the symplectization of standard contact $(2n-1)$-space with Legendrian boundary $\Sigma$ can be glued to itself along $\Sigma$. This gives a Legendrian embedding $\Lambda(L,L)$ of the double of $L$…