Lengths of Reeb chords and Viterbo restriction

Abstract

Let $\Lambda$ be a Legendrian in the contact boundary of a Liouville domain $\Omega$. We explain how the non-existence of Reeb chords with endpoints on $\Lambda$ of length up to $a$ enables one to embed $D_{\epsilon}T^{*}\Lambda\times D(a)$ into $\Omega$ in an exact way. As in earlier work of Zhengyi Zhou, we use the Viterbo restriction map to deduce a contradiction in certain cases. In particular, we show that if $M$ admits a submersion from a product of spheres (e.g., the $n$-torus), then all compact Legendrians $\Lambda\subset ST^{*}M$ admit a Reeb chord for every choice of contact form $ST^{*}M$. The obstruction we use in this case is based on the idea of inverting the degree-$n$ classes in cohomology, and is similar to the notion of string point invertibility introduced by Egor Shelukhin.