For every tame contact manifold (Y,xi), every contact form alpha with ker alpha=xi, and every compact Legendrian submanifold Lambda⊂ Y, there exist T>0 and a smooth Reeb trajectory gamma colon[0,T]→ Y such that dotgamma(t)=R_alpha(gamma(t)) for all t∈[0,T], gamma(0),gamma(T)∈Lambda, and gamma is non-constant.

Arnol'd's chord conjecture for closed Legendrians

Arnol'd’s chord conjecture asserts that every compact Legendrian submanifold in a tame contact manifold has a Reeb chord. A December 2024 preprint claims this in full generality, but the supplied evidence does not independently validate the claim.

Known results

Mohnke, 2001: the standard-contact $S^3$ case.
Hutchings–Taubes: the conjecture in contact dimension $3$ .
Conormal bundles, 2024: all conormal Legendrians, and Legendrians isotopic to them, in $S^*M$ for closed $M$ .
Various filling and topological hypotheses, 2020, including vanishing symplectic cohomology and certain $K(\pi,1)$ Legendrians.

August 2026 expansion and outstanding full-proof claim

Egor Shelukhin’s August 2026 preprint is reported to prove the conjecture for a broad new class, extending results for examples such as $ST^*T^3$ and related contact manifolds. This is progress, not a general resolution. The December 2024 contact-instanton paper nevertheless claims a complete proof for every tame contact manifold and compact Legendrian; no independent verification, referee report, or retraction was found.

Current status (as of August 2026): many substantial classes are settled and a 2024 preprint claims the full conjecture, but that general proof remains unverified.

Arnol'd's chord conjecture for closed Legendrians

Progress summary

Primary source

Additional references

Solutions 0