Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1808.08459

Chekanov's dichotomy in contact topology

Daniel Rosen·Jun Zhang
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we study submanifolds of contact manifolds. The main submanifolds we are interested in are contact coisotropic submanifolds. Based on a correspondence between symplectic and contact coisotropic submanifolds, we can show contact coisotropic submanifolds admit a $C^0$-rigidity, similar to Humilière-Leclercq-Seyfaddini's coisotropic rigidity on symplectic manifolds. Moreover, based on Shelukhin's norm defined on the contactomorphism group, we define a Chekanov type pseudo-metric on the orbit space of a fixed submanifold of a contact manifold. Moreover, we can show a dichotomy of (non-)degeneracy of this pseudo-metric when the dimension of this fixed submanifold is equal to the one for a Legendrian submanifold. This can be viewed as a contact topology analogue to Chekanov's dichotomy of (non-)degeneracy of Chekanov-Hofer's metric on the orbit space of a Lagrangian submanifold.

Explore related subjects

math.SGmath.GT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Daniel Rosen, Jun Zhang. 2021-07-24. Chekanov's dichotomy in contact topology. https://doi.org/10.4310/mrl.2020.v27.n4.a9

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Surjectivity of real-linear Cauchy--Riemann operators: from the minimal Harder--Narasimhan slope to automatic transversality

This paper relates the minimal Harder--Narasimhan slope to the surjectivity of real-linear Cauchy--Riemann operators. We establish a conformally invariant $L^2$ criterion and an asymptotic slope criterion, which yield higher-rank automatic transversality criteria for pseudoholomorphic curves beyond the classical rank-one framework. Applications to pseudoholomorphic spheres in $S^6$ provide quantitative $L^2$ obstructions to the integrability of almost complex structures.

math.SG↗
arXiv2026

Welschinger invariants and the Conway polynomial

Welschinger showed that counts of connected holomorphic disks with Lagrangian boundary in symplectic 6-manifolds, meeting at least one boundary constraint, can be made invariant by correcting them with counts of disconnected disks weighted by certain "self-linking" numbers. We show his invariant is the lowest order term in an all-genus curve count where curves are weighted by the Conway polynomials of their boundaries. This in turn is a specialization of the skein-valued curve count, but can be defined without the 4-chain and vector field used in that setup.

math.SG↗
arXiv2026

KAM splittings and equidistributed periodic orbits for stable hypersurfaces

We show that any stable hypersurface of a symplectic $4$-manifold, on which the cohomology class of the symplectic form restricts to a multiple of a rational class, can be $C^\infty$-approximated by (possibly unstable) hypersurfaces whose closed characteristics equidistribute. The cohomological condition is necessary due to a famous example of Herman. The proof combines KAM theory with recent quantitative closing lemmas for Reeb flows and area-preserving maps. As a further application, we prove that every geodesible volume-preserving vector field on a closed three-manifold can be $C^\infty$-approximated by volume-preserving vector fields with equidistributed periodic orbits.

math.SG↗