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arxiv: 2606.23663 · v1 · pith:BBYVJ4SHnew · submitted 2026-06-22 · 🧮 math.SG

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

Dylan Cant This is my paper

Pith reviewed 2026-06-26 05:31 UTC · model grok-4.3

classification 🧮 math.SG
keywords Arnold chord conjectureLegendrian submanifoldsReeb chordsGutt-Hutchings capacitiesLiouville domainsuniformly convex domainssymplectic capacitiescontact geometry
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The pith

Any E3 Legendrian on the boundary of a Liouville domain bounds a Reeb chord of length at most liminf c_k(Ω)/k.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a structural result for Legendrians that are eventually equivariantly essential, meaning the Gutt-Hutchings capacities of their cotangent bundles become infinite for large k. It shows that any such Legendrian sitting in the boundary of a Liouville domain must bound a Reeb chord whose length is controlled by the liminf of the domain's normalized Gutt-Hutchings capacities. The authors apply this general fact to the boundaries of uniformly convex domains in four-dimensional space, using capacity information transferred from ellipsoid embeddings, and thereby establish the strong Arnold chord conjecture in this case. A sympathetic reader cares because the result guarantees the existence of short chords for essential Legendrians on contact three-spheres, connecting to the dynamics of the Reeb flow.

Core claim

We prove a structural result for Legendrians L which are eventually equivariantly essential (E3), in the sense that the kth 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. This is used to prove the strong Arnol'd chord conjecture for the boundary of a uniformly convex domain in R^4.

What carries the argument

The E3 condition on Legendrians, which ensures the Gutt-Hutchings capacities of the cotangent bundle become infinite for large k and thereby permits a chord-length upper bound in terms of the ambient domain's capacities.

If this is right

  • The strong Arnol'd chord conjecture holds for the boundary of every uniformly convex domain in R^4.
  • Every E3 Legendrian in the boundary of any Liouville domain bounds a Reeb chord whose length is bounded by the liminf of the normalized capacities.
  • The chord-length bound is available in the general Liouville setting whenever the E3 condition is satisfied.
  • Capacity data from ellipsoid embeddings transfers to give explicit length controls on the convex domains.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If analogous embedding constructions exist for other families of domains, the same chord bound would extend beyond the uniformly convex case.
  • The growth rate of capacities could determine chord lengths for E3 Legendrians in higher-dimensional or non-convex Liouville domains.
  • Direct computation of capacity sequences on explicit convex examples would allow numerical checks of the predicted chord lengths.

Load-bearing premise

An ellipsoid embedding construction applies directly to uniformly convex domains in four dimensions and supplies the capacity control needed to invoke the structural result for E3 Legendrians.

What would settle it

A uniformly convex domain in R^4 together with an E3 Legendrian on its boundary that has no Reeb chord of length at most liminf c_k(Ω)/k would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.23663 by Dylan Cant.

Figure 1
Figure 1. Figure 1: Homological conventions. A.2. Borel equivariant data. We follow the ideas of [SS10] for G = Z/2Z, which mostly adapt easily to the case of G = S 1 . A discussion of this case is given in [BO17, §2.2] and [Zha16, §2.3] [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
read the original 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 $\Omega$ bounds a chord of length at most $\liminf c_{k}(\Omega)/k$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proves the strong Arnol'd chord conjecture for the boundary of a uniformly convex domain in R^4. It first establishes a general structural result: any Legendrian L that is eventually equivariantly essential (E3), meaning c_k(D^*TL) = ∞ for all sufficiently large k, bounds a Reeb chord of length at most liminf_k c_k(Ω)/k when embedded in the boundary of a Liouville domain Ω. This is then applied to uniformly convex domains by invoking an ellipsoid embedding construction of Edtmair together with prior results of Kang-Zhang.

Significance. If the central claims hold, the work would constitute a meaningful advance in symplectic and contact geometry by confirming a strong form of the Arnol'd chord conjecture in dimension 4 for an important class of domains. The structural result for E3 Legendrians is potentially reusable and the bound is expressed directly in terms of the Gutt-Hutchings capacities without additional fitted parameters.

major comments (2)
  1. [Abstract] Abstract, paragraph 2: the claim that Edtmair's ellipsoid embedding construction 'applies directly' to uniformly convex domains in R^4 and 'allows control' of the Gutt-Hutchings capacities c_k(Ω) is load-bearing for the second pillar of the argument. The manuscript must supply the explicit embedding, the induced filtration on symplectic homology, and the verification that the resulting capacity sequence yields a finite liminf while preserving the E3 property for the relevant Legendrian; without these steps the implication to the strong Arnol'd conjecture does not follow.
  2. [Structural result section] The structural result for E3 Legendrians (the inequality length ≤ liminf c_k(Ω)/k) relies on the definition of E3 via the capacities of the cotangent bundle D^*TL. Any hidden dependence on the choice of Liouville form or on the Reeb flow outside a neighborhood of L must be checked, as it would affect whether the bound is uniform for all uniformly convex domains.
minor comments (1)
  1. Notation for the Gutt-Hutchings capacities c_k should be introduced with a brief reminder of their definition or a reference to the precise normalization used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph 2: the claim that Edtmair's ellipsoid embedding construction 'applies directly' to uniformly convex domains in R^4 and 'allows control' of the Gutt-Hutchings capacities c_k(Ω) is load-bearing for the second pillar of the argument. The manuscript must supply the explicit embedding, the induced filtration on symplectic homology, and the verification that the resulting capacity sequence yields a finite liminf while preserving the E3 property for the relevant Legendrian; without these steps the implication to the strong Arnol'd conjecture does not follow.

    Authors: We agree that the manuscript would benefit from additional explicit details on this step. In the revised version we will insert a new subsection that (i) recalls the precise statement of Edtmair's ellipsoid embedding for uniformly convex domains in R^4, (ii) describes the induced filtration on the symplectic homology of the target domain, and (iii) verifies that the resulting Gutt-Hutchings capacity sequence satisfies liminf c_k(Ω)/k < ∞ while the E3 property of the Legendrian is preserved. This will make the passage from the structural result to the strong Arnol'd conjecture fully explicit. revision: yes

  2. Referee: [Structural result section] The structural result for E3 Legendrians (the inequality length ≤ liminf c_k(Ω)/k) relies on the definition of E3 via the capacities of the cotangent bundle D^*TL. Any hidden dependence on the choice of Liouville form or on the Reeb flow outside a neighborhood of L must be checked, as it would affect whether the bound is uniform for all uniformly convex domains.

    Authors: The E3 condition is defined intrinsically via the Gutt-Hutchings capacities of D^*TL and is therefore independent of any choice of Liouville form on the ambient domain Ω. The proof of the chord-length bound uses only the filtered symplectic homology of Ω together with the embedding of L; it does not invoke the Reeb dynamics of Ω away from a neighborhood of L. We will add a short clarifying paragraph immediately after the statement of the structural result that records this independence and confirms uniformity of the bound across all uniformly convex domains. revision: yes

Circularity Check

0 steps flagged

No significant circularity; proof chain relies on independent external results

full rationale

The paper establishes a general structural result for E3 Legendrians (any such L in ∂Ω bounds a chord of length ≤ liminf c_k(Ω)/k) and invokes the Edtmair ellipsoid embedding plus Kang-Zhang ideas only as external inputs to handle the uniformly convex case in R^4. No equation or claim inside the paper reduces a derived quantity to a fitted parameter or self-defined input by construction, nor does any load-bearing step collapse to a self-citation chain. The cited constructions are treated as independent and are not reproduced or redefined within the manuscript, satisfying the criteria for a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of Gutt-Hutchings capacities and the applicability of a cited embedding construction; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Standard properties of Gutt-Hutchings capacities in symplectic homology
    Invoked to define E3 and the liminf bound.
  • domain assumption Edtmair's ellipsoid embedding construction applies to uniformly convex domains
    Used to control capacities for the specific case in R^4.

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