The strong Arnol'd chord conjecture for the boundary of a uniformly convex domain in $\mathbb{R}^{4}$

Abstract

Following the idea of Jungsoo Kang and Jun Zhang, we prove the strong Arnol'd chord conjecture for the boundary of a uniformly convex domain in $\mathbb{R}^{4}$, using an ellipsoid embedding construction due to Oliver Edtmair. We prove a general structural result for Legendrians $L$ which are eventually equivariantly essential (E3), in the sense that the $k$th Gutt-Hutchings capacity $c_{k}(D^{*}TL)$ is infinite for $k$ large enough. We show that any E3 Legendrian in the boundary of a Liouville domain $Ω$ bounds a chord of length at most $\liminf c_{k}(Ω)/k$.