Arnol’d’s chord conjecture for compact contact manifolds

Prove that every compact Legendrian submanifold in every compact contact manifold admits a Reeb chord.

Background

The paper explains that its algebraic examples were motivated by Arnol’d’s chord conjecture in symplectic topology. The conjecture is known in several cases, including boundaries of subcritical Weinstein domains, Weinstein domains with vanishing symplectic cohomology, and all three-dimensional contact manifolds, but remains unresolved in general.

References

This conjecture is known for boundaries of Weinstein domains which are subcritical or more generally have vanishing symplectic cohomology ; it is also known for all three dimensional contact manifolds . There has been a spate of recent progress , but the conjecture remains wide open in general.

— A proper dg algebra which does not cogenerate  (2609.34193 - Hu et al., 28 Sep 2026) in Section 1, Introduction; Section 7, Section \ref{sec: chord}