On the strong Arnold chord conjecture for convex contact forms

TL;DR

The paper addresses the strong Arnold chord conjecture for convex contact forms on the standard sphere by proving existence of Reeb chords with distinct endpoints for Legendrian submanifolds when the boundary of a strictly convex domain has Morse-Bott minimal Reeb orbits. The approach leverages Rabinowitz Floer homology and symplectic homology capacities to produce Floer cylinders from minimal periodic orbits to Reeb chords, yielding chords with strictly lower action than the minimal orbit. A non-convex counterexample demonstrates the necessity of convexity for the strict bound. The work also establishes that c_RF H(∂W) = c_SH(W) = A_min(∂W) in the convex setting and links these capacities to the strong chord conjecture via a closed-open map and Morse-Bott perturbations, extending prior results and introducing a Morse-Bott relaxation of nondegeneracy assumptions.

Abstract

The original Arnold chord conjecture states that every closed Legendrian submanifold of the standard contact sphere $S^{2n-1}$ admits a Reeb chord with distinct endpoints with respect to any contact form. In this paper, we prove this conjecture for contact forms induced by strictly convex embeddings into $\mathbb{R}^{2n}$ under the assumption that minimal periodic Reeb orbits are of Morse-Bott type. We also provide a counterexample when the convexity condition is not satisfied.

Figures (3)

• Figure 2: graphs of $h_{a}^\mathrm{lin}$ and $h_a$
• Figure 3: Curves in $\mathcal{N}$
• Figure 4: Possible limit curves (i) and (ii)

Theorems & Definitions (5)

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