Lengths of Reeb chords and Viterbo restriction

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 $D_εT^{*}Λ\times D(a)$ 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 $M$ is covered by a product of spheres (e.g., the $n$-torus), then all compact Legendrians $Λ\subset ST<sup>{*}M$ admit a Reeb chord for every choice of contact form $ST<sup>{*}M$. 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.

• The paper introduces the n-invertibility capacity, a filtered symplectic-cohomology invariant that gives every compact Legendrian in a Liouville domain a Reeb chord of period at most the capacity when it is finite.
• The authors combine a chord-free Legendrian embedding, structure-preserving filtered Viterbo restriction, and a new filtered Künneth map to turn algebraic relations in symplectic cohomology into quantitative geometric bounds.
• For unit cotangent bundles whose bases are finitely covered by products of spheres, including tori, the method proves Reeb chord existence for every contact form and yields rationality bounds for closed aspherical Lagrangians in T*T^n.

Overview

The paper by Broćić and Cant establishes quantitative bounds on the lengths of Reeb chords of Legendrian submanifolds in Liouville domains, using symplectic cohomology together with the Viterbo restriction map. The central object introduced is the $n$-invertibility capacity $c_{\mathrm{ni}}(\Omega)$, a filtered refinement of an algebraic condition on $\mathit{SH}(W)$ involving the PSS map, the BV operator $\Delta$, and the pair-of-pants product. The main theorem states that every compact Legendrian $\Lambda \subset \Omega$ admits a non-constant Reeb chord of period at most $c_{\mathrm{ni}}(\Omega)$ whenever this capacity is finite. As applications, the authors prove that every compact Legendrian in $ST^{*}M$ has a Reeb chord for every contact form when $M$ is finitely covered by a product of spheres (in particular for the $n$-torus), and they bound the rationality constants of closed aspherical Lagrangians in $T^{*}T^{n}$.

The notion of $c_{\mathrm{ni}}(\Omega)$0-invertibility

A Liouville manifold $c_{\mathrm{ni}}(\Omega)$1 is defined to be $c_{\mathrm{ni}}(\Omega)$2-invertible if the only $c_{\mathrm{ni}}(\Omega)$3-invariant ideal of $c_{\mathrm{ni}}(\Omega)$4 containing $c_{\mathrm{ni}}(\Omega)$5 is all of $c_{\mathrm{ni}}(\Omega)$6. This condition is closely related to Shelukhin's string point invertibility. To quantify it, the authors work with the persistence module $c_{\mathrm{ni}}(\Omega)$7 of Floer cohomologies of Hamiltonians agreeing with $c_{\mathrm{ni}}(\Omega)$8 at infinity, and define a filtration of ideals $c_{\mathrm{ni}}(\Omega)$9 generated iteratively from $\mathit{SH}(W)$0 and applications of $\mathit{SH}(W)$1. The colimit $\mathit{SH}(W)$2 is shown to be the smallest $\mathit{SH}(W)$3-invariant ideal containing $\mathit{SH}(W)$4, and the capacity

$\mathit{SH}(W)$5

is finite exactly when $\mathit{SH}(W)$6 is $\mathit{SH}(W)$7-invertible. Two classes of examples are identified: domains with vanishing symplectic cohomology, which recover a result of Zhou, and cotangent bundles of products of spheres, treated below.

From chord-free Legendrians to exact embeddings

The first step of the proof is a geometric construction in the spirit of Mohnke and Opshtein, formulated as a version of the 1-jet neighborhood theorem. If a Legendrian isotopy $\mathit{SH}(W)$8 satisfies $\mathit{SH}(W)$9 along every smooth path and the $\Delta$0 are pairwise disjoint, then there is an exact embedding $\Delta$1. In particular, absence of Reeb chords of length up to $\Delta$2 yields such an embedding.

The second step is the Viterbo restriction map $\Delta$3 associated to an exact embedding of Liouville domains, proven here in a filtered form compatible with the PSS map, BV operator, and product. If $\Delta$4, choosing $\Delta$5 forces the unit to lie in $\Delta$6, which vanishes because the pullback $\Delta$7 is zero — there are no degree-$\Delta$8 classes on $\Delta$9. A third ingredient, a filtered Künneth map $\Lambda \subset \Omega$0 for a Liouville domain $\Lambda \subset \Omega$1 times a disk $\Lambda \subset \Omega$2, shows that vanishing of the unit descends to $\Lambda \subset \Omega$3, contradicting the Abouzaid comparison with string topology, under which the unit cannot vanish. This contradiction proves the main theorem.

The construction of the filtered Künneth map is one of the paper's more delicate technical contributions. Standard Künneth isomorphisms identify $\Lambda \subset \Omega$4, but since $\Lambda \subset \Omega$5 these are useless here; instead the authors construct admissible Hamiltonians of the form $\Lambda \subset \Omega$6 on $\Lambda \subset \Omega$7, with a carefully chosen almost complex structure that is Liouville-equivariant outside a rectangle and genuinely split near $\Lambda \subset \Omega$8. A compactness argument shows all Floer cylinders and PSS solutions remain in $\Lambda \subset \Omega$9, giving a canonical isomorphism preserving the unit; the requirement $c_{\mathrm{ni}}(\Omega)$0 is essential for the Fredholm argument via the operator $c_{\mathrm{ni}}(\Omega)$1.

Products of spheres and the Arnol'd chord conjecture

The key topological input is the construction, entirely within string topology, of explicit relations in $c_{\mathrm{ni}}(\Omega)$2 for $c_{\mathrm{ni}}(\Omega)$3. For a single sphere, the authors build classes $c_{\mathrm{ni}}(\Omega)$4 (the rotation action class), $c_{\mathrm{ni}}(\Omega)$5 obtained by applying a "completing manifolds" operation to a generalized section $c_{\mathrm{ni}}(\Omega)$6, and prove

$c_{\mathrm{ni}}(\Omega)$7

The generalized section exists even though $c_{\mathrm{ni}}(\Omega)$8 admits no section for even $c_{\mathrm{ni}}(\Omega)$9: after the unoriented blow-down $ST^{*}M$0, the push-forward of a constant vector field extends smoothly. The proof that $ST^{*}M$1, where $ST^{*}M$2 is the class induced by a point of $ST^{*}M$3, proceeds by an explicit homotopy between $ST^{*}M$4 and the open-book class $ST^{*}M$5 of earlier work. Cartesian products then yield classes $ST^{*}M$6 with $ST^{*}M$7, proving $ST^{*}M$8 is $ST^{*}M$9-invertible via the BV-algebra morphism $M$0.

Consequently, any compact Legendrian in $M$1, for $M$2 finitely covered by a product of spheres, admits a Reeb chord for every contact form. This solves the Arnol'd chord conjecture for this class of ambient manifolds. Notably, the result applies to $M$3, an aspherical case where neither higher dilation arguments nor orientation tricks (which require rationally inessential or $M$4-nontrivial manifolds respectively) apply. The method also handles uniformly fast time-dependent Reeb flows, which bubbling-based approaches do not readily accommodate.

Rationality constants of aspherical Lagrangians in $M$5

The second main application concerns the rationality constant $M$6 of a closed aspherical Lagrangian $M$7 in the unit codisk bundle of $M$8: either $M$9 is exact or $n$0 for a constant depending only on $n$1. Since a non-exact Lagrangian does not admit an exact Weinstein neighborhood, the standard Viterbo restriction does not apply directly; the authors use Zhou's truncated Viterbo restriction map, proven here with compatibility with the BV operator, product, and free homotopy class decompositions, valid when $n$2.

Applying this map to invertible classes $n$3 with $n$4 in the homotopy classes of $n$5 produces non-zero classes $n$6 of degree at least $n$7 in specific free homotopy classes. Two lemmas of Latschev–Oancea then imply that powers of the corresponding elements of $n$8 generate a finite-index subgroup isomorphic to $n$9 lying in the kernel of $T^{*}T^{n}$0. Since $T^{*}T^{n}$1 is aspherical, the associated cover is homotopy equivalent to $T^{*}T^{n}$2 and hence closed, so the covering is finite; but the kernel of a non-zero homomorphism to $T^{*}T^{n}$3 has infinite index. Therefore $T^{*}T^{n}$4, i.e., $T^{*}T^{n}$5 is exact. The authors note that previous results on aspherical Lagrangians do not apply when the ambient space is $T^{*}T^{n}$6, making this a genuinely new instance of the rationality bound.

Relation to prior work

The paper situates itself against several strands. Zhou's approach via higher dilations yields chords for aspherical Legendrians in unit cotangent bundles of rationally inessential manifolds, using $T^{*}T^{n}$7-equivariant symplectic homology and Goodwillie's theorem; the present method is complementary, applying to aspherical bases like $T^{*}T^{n}$8 where dilation classes do not exist. Orientation-sensitive arguments give chords for spin Legendrians in $T^{*}T^{n}$9 when $c_{\mathrm{ni}}(\Omega)$00 pairs non-trivially with $c_{\mathrm{ni}}(\Omega)$01, e.g., $c_{\mathrm{ni}}(\Omega)$02, since $c_{\mathrm{ni}}(\Omega)$03 in those cases. The strong Arnol'd chord conjecture (chords with distinct endpoints) remains out of reach of these methods because the length bound always dominates the systole of $c_{\mathrm{ni}}(\Omega)$04.

Limitations and open questions

Several restrictions are acknowledged explicitly. The chord length bound requires finiteness of $c_{\mathrm{ni}}(\Omega)$05, and the authors observe that for $c_{\mathrm{ni}}(\Omega)$06 of negative sectional curvature there appear to be too few relations between top-degree PSS classes and other classes for the method to apply; whether a symplectic-cohomological upper bound exists for Legendrian knots in domains over $c_{\mathrm{ni}}(\Omega)$07 ($c_{\mathrm{ni}}(\Omega)$08) is left open, though chord existence itself is known there by Hutchings–Taubes. The spin hypothesis in the orientation-based claim is not known to be removable without further conditions on the inclusion $c_{\mathrm{ni}}(\Omega)$09. The strong Arnol'd conjecture cannot be addressed by this framework for the systole reason above. Finally, the filtered Künneth construction and the Viterbo restriction machinery rely on characteristic-two coefficients throughout the body of the paper, and the canonicity discussion deliberately fixes the almost complex structure rather than treating it as fully auxiliary data.

Conclusion

This paper converts an algebraic invertibility property of symplectic cohomology — packaged as a computable capacity $c_{\mathrm{ni}}(\Omega)$10 — into a sharp existence statement for short Reeb chords, via the combination of a stabilized codisk-bundle embedding theorem, structure-compatible filtered Viterbo restriction, and a new filtered Künneth isomorphism. The resulting chord-existence theorem for $c_{\mathrm{ni}}(\Omega)$11 with $c_{\mathrm{ni}}(\Omega)$12 covered by products of spheres, including the torus, and the rationality bound for aspherical Lagrangians in $c_{\mathrm{ni}}(\Omega)$13, extend the reach of Viterbo-restriction methods to settings inaccessible to dilation and orientation techniques.