Lengths of Reeb chords and Viterbo restriction

2026/06/15 by Filip Broćić, Dylan Cant · 1 voice · 2 citations
Mathematics · #math.SG

paper · pdf

arxiv published 2026/06/15 · arxiv updated 2026/07/06

Abstract

Let Λ be a Legendrian in the contact boundary of a Liouville domain Ω. We explain how the non-existence of Reeb chords with endpoints on Λ of length up to a enables one to embed D_εT^*Λ× D(a) into Ω 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 Λ⊂ 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.

Cited by

• The strong Arnol'd chord conjecture for the boundary of a uniformly convex domain in ℝ⁴
• The Arnol'd chord conjecture for ST^*T³

Discussions

• Filip Bro\'ci\'c, Dylan Cant: Lengths of Reeb chords and Viterbo restriction https://arxiv.org/abs/2606.16270 https://arxiv.org/pdf/2606.16270 https://arxiv.org/html/2606.16270 [bsky, 0 points, 0 comments]

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