arxiv: 2604.19237 · v1 · submitted 2026-04-21 · 🧮 math.SG · math.DS

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Bott-integrable contact forms with large systolic ratio

Hansj\"org Geiges , Jakob Hedicke , Murat Sa\u{g}lam

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Pith reviewed 2026-05-10 01:25 UTC · model grok-4.3

classification 🧮 math.SG math.DS

keywords Bott-integrable contact formssystolic ratioLutz formscontact plugs3-manifoldscontact geometry

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The pith

Bott-integrable contact forms on closed 3-manifolds have no universal upper bound on their systolic ratio.

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

The paper establishes that the systolic ratio of Bott-integrable contact forms on closed 3-manifolds has no finite upper bound that works for all such forms. A sympathetic reader would care because the result indicates that the integrability condition leaves room for substantial geometric variation rather than forcing a uniform cap on this ratio. The argument proceeds by replacing Lutz forms with piecewise linear approximations and inserting a plug that raises the ratio. The constructions are carried out so that Bott-integrability is retained throughout.

Core claim

There is no universal upper bound for the systolic ratio of Bott-integrable contact forms on closed 3-manifolds. For the proof, we study piecewise linear approximations of Lutz forms and establish integrability of a plug constructed by Abbondandolo, Bramham, Hryniewicz and Salomão for pushing up the systolic ratio.

What carries the argument

Piecewise linear approximations of Lutz forms combined with the Abbondandolo-Bramham-Hryniewicz-Salomão plug, which together preserve Bott-integrability while allowing the systolic ratio to increase without bound.

If this is right

The systolic ratio can be made larger than any given positive number by suitable choice of the Bott-integrable form.
Integrability of the contact form does not impose an upper limit on the systolic ratio.
The same conclusion holds on every closed 3-manifold.
The constructions supply concrete examples witnessing the absence of a universal bound.

Where Pith is reading between the lines

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

The same plug technique might be applied to produce examples with large systolic ratio while controlling other dynamical features of the Reeb flow.
It would be natural to ask whether the absence of an upper bound persists when the contact form is required to satisfy stronger integrability conditions.
Quantitative versions of the result could track how rapidly the ratio grows with the number of pieces in the linear approximation.

Load-bearing premise

The piecewise linear approximations of Lutz forms and the given plug both preserve Bott-integrability on the resulting contact forms.

What would settle it

An explicit closed 3-manifold together with a fixed constant C such that every Bott-integrable contact form on the manifold has systolic ratio at most C.

Figures

Figures reproduced from arXiv: 2604.19237 by Hansj\"org Geiges, Jakob Hedicke, Murat Sa\u{g}lam.

**Figure 1.** Figure 1: The smoothened piecewise linear Lutz form. Proposition 3 (ABHS). Let r0, δ > 0 and a primitive λ of the standard area form dx∧dy on r0D be given. Then there is a contact form β = βr0,λ,δ on the solid torus r0D × R/Z with the following properties: (i) β = λ+ ds near the boundary of the solid torus; in particular, the Reeb vector field equals ∂s near the boundary. (ii) β is isotopic relative to a neighbourho… view at source ↗

**Figure 2.** Figure 2: The Morse foliation on the annulus An ⊂ D/Zn. with πr2 ℓ = (1 − δ) area(Aℓ). This extends to an embedding Ψℓ : rℓD × R/Z −→ Aℓ × (R/Z)x1 = Wℓ [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: The Morse foliation on the annulus Aℓ = R/Z × [−1, 1]. Here the summand δ corresponds to the volume of the image of Ψℓ after the modification of βε into βε,δ and comes from part (iv) of Proposition 3; the summand 2aℓbℓδ is the volume of the complement of this image, where βε is left unmodified. The number of summands (i.e., the range of ℓ) and the aℓ, bℓ depend on the chosen ε > 0. By then choosing δ > 0 … view at source ↗

read the original abstract

We show that there is no universal upper bound for the systolic ratio of Bott-integrable contact forms on closed 3-manifolds, thus providing further evidence for the relative flexibility of integrable contact forms. For the proof, we study piecewise linear approximations of Lutz forms and establish integrability of a `plug' constructed by Abbondandolo, Bramham, Hryniewicz and Salom\~ao for pushing up the systolic ratio.

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.

Desk Editor's Note private letter to a colleague

Bott-integrable contact forms on closed 3-manifolds have unbounded systolic ratios, shown by piecewise-linear Lutz approximations plus an integrable ABHS plug.

read the letter

The one thing to know is that this paper shows there is no universal upper bound for the systolic ratio of Bott-integrable contact forms on closed 3-manifolds. They do this by approximating Lutz forms with piecewise linear ones and inserting the ABHS plug, which they prove remains integrable. What is new is the unboundedness result specifically for the Bott-integrable class, along with the verification that the plug can be made to work with these approximations. The paper does a good job laying out how the piecewise linear versions behave and confirming the integrability condition for the plug. The potential soft spot is in the details of how the Bott function is preserved when gluing in the plug. Since Bott-integrability requires a smooth function constant on periodic orbits, any mismatch at the boundaries could be a problem. The construction seems to address this by design, but it is the step that needs the most careful reading to confirm there are no hidden issues with periodicity or invariance. This work is aimed at researchers in symplectic and contact geometry who care about systolic ratios and integrable structures. It provides further evidence that integrability does not force a bound on the ratio, which is useful for understanding flexibility in this setting. I would recommend sending it to peer review. The result is substantive enough and the method is constructive, so it merits a full referee process even if some revisions are needed on the technical checks.

Referee Report

2 major / 2 minor

Summary. The paper claims there is no universal upper bound on the systolic ratio of Bott-integrable contact forms on closed 3-manifolds. The argument constructs examples by taking piecewise-linear approximations to Lutz forms on the 3-sphere (or other manifolds) and inserting the Abbondandolo-Bramham-Hryniewicz-Salomão plug in a manner asserted to preserve Bott-integrability, thereby driving the systolic ratio arbitrarily large.

Significance. If the constructions are verified, the result supplies concrete evidence that Bott-integrable contact forms are more flexible than general contact forms with respect to systolic inequalities. It directly extends the ABHS plug technique and the study of Lutz forms, furnishing a family of examples with unbounded systolic ratio while maintaining the Bott condition.

major comments (2)

[plug-insertion construction (likely §3–4)] The central step—preservation of Bott-integrability under plug insertion—requires an explicit global Bott integral I after gluing. The manuscript must construct I on the plug region, verify that dI(R)=0 everywhere (including at the interface), and confirm that the critical set remains a union of compact invariant submanifolds with periodic Reeb flow. Without these verifications the claim that the modified form remains Bott-integrable is not load-bearing for the unbounded-ratio theorem.
[piecewise-linear approximation (likely §2)] For the piecewise-linear approximations of Lutz forms, the manuscript must show that the approximating 1-form admits a Bott integral whose critical set consists of compact invariant submanifolds. It is unclear whether the linear pieces preserve the required periodicity and invariance conditions near the approximation loci; a concrete check or explicit I for the approximated form is needed before the plug can be inserted.

minor comments (2)

[Introduction] The notation for the systolic ratio (and its normalization) should be stated once in the introduction with a precise formula, rather than relying on references to prior work.
[Construction overview] A short diagram or schematic of the gluing region and the support of the plug would clarify how the local model interfaces with the global Lutz approximation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comments. The points raised concern the need for more explicit verifications of Bott-integrability in both the plug insertion and the piecewise-linear approximations. We will revise the manuscript to supply these details, thereby making the arguments fully rigorous.

read point-by-point responses

Referee: [plug-insertion construction (likely §3–4)] The central step—preservation of Bott-integrability under plug insertion—requires an explicit global Bott integral I after gluing. The manuscript must construct I on the plug region, verify that dI(R)=0 everywhere (including at the interface), and confirm that the critical set remains a union of compact invariant submanifolds with periodic Reeb flow. Without these verifications the claim that the modified form remains Bott-integrable is not load-bearing for the unbounded-ratio theorem.

Authors: We agree that an explicit global Bott integral is required for the claim to be load-bearing. In the revised manuscript we will construct I on the plug region by extending the Bott integral of the original Lutz form using the integrable structure built into the ABHS plug (which is known to admit a Bott integral with the required critical-set properties). We will verify dI(R)=0 at the gluing interface by direct computation of the matching conditions on the defining functions, and we will confirm that the critical set remains a union of compact invariant submanifolds carrying periodic Reeb flow, since the plug only alters dynamics inside a region whose boundary orbits are already periodic and invariant. revision: yes
Referee: [piecewise-linear approximation (likely §2)] For the piecewise-linear approximations of Lutz forms, the manuscript must show that the approximating 1-form admits a Bott integral whose critical set consists of compact invariant submanifolds. It is unclear whether the linear pieces preserve the required periodicity and invariance conditions near the approximation loci; a concrete check or explicit I for the approximated form is needed before the plug can be inserted.

Authors: We accept that a concrete verification is necessary. In the revision we will exhibit an explicit Bott integral I for each piecewise-linear approximation of the Lutz form. The integral will be defined piecewise, constant along the Reeb vector field on each linear piece, and we will check that the critical sets are unions of compact invariant submanifolds (circles or tori) on which the Reeb flow is periodic. Near the approximation loci we will verify invariance and periodicity by direct comparison with the original Lutz form, ensuring the conditions hold before the plug is inserted. revision: yes

Circularity Check

0 steps flagged

No circularity; explicit construction and verification of Bott-integrability are independent of the target claim.

full rationale

The paper's central result follows from studying piecewise-linear approximations to Lutz forms, inserting the ABHS plug (from independent prior work by Abbondandolo-Bramham-Hryniewicz-Salomão), and establishing that the composite remains Bott-integrable. No step reduces by definition, fitted-parameter renaming, or self-citation chain to the systolic-ratio bound itself; the verification of the global integral I and interface conditions is presented as new content within the manuscript rather than presupposed. This is the standard pattern for a constructive existence proof in contact geometry and yields no load-bearing circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof draws on standard differential geometry and contact topology without introducing free parameters or new postulated entities.

axioms (1)

standard math Standard axioms of smooth manifold theory, contact structures, and Reeb dynamics
Invoked throughout the study of contact forms and integrability on closed 3-manifolds.

pith-pipeline@v0.9.0 · 5360 in / 1024 out tokens · 35559 ms · 2026-05-10T01:25:03.709979+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 1 canonical work pages

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