Tightness of Chekanov's bound on displacement energy for some Lagrangian knots

David Keren Yaar

REVIEW 3 major objections 4 minor 5 references

Tightness of Chekanov's bound on displacement energy for some Lagrangian knots

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For small Chekanov tori in $\mathbb{C}P^n$ and monotone Brendel tori in $\mathbb{C}^3$, the displacement energy equals the smallest holomorphic disk area, so Chekanov's bound is tight.

desk verdict A solid Chekanov-torus computation paired with a load-bearing but fixable gap in the Brendel-torus calculation; worth sending to a referee. read the letter →

arxiv 2507.21282 v1 pith:ZKLHOE4X submitted 2025-07-28 math.SG

David Keren Yaar This is my paper · ORCID

classification math.SG MSC 53D1253D20

keywords displacementenergyJ-holomorphicdisksLagrangiantoriChekanovtorusBrendelHofernormMaslovindex

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper examines Chekanov's lower bound on displacement energy: the displacement energy $e(L)$ of a Lagrangian submanifold is always at least $\hbar(L)$, the smallest area of a pseudo-holomorphic disk with boundary on $L$. The paper proves that the bound is tight for two families of Lagrangian tori: small Chekanov tori $T^n_{\mathrm{PrCh}}(a)\subset \mathbb{C}P^n$ with $0

What carries the argument

Two mechanisms carry the argument. For the upper bound on $e$, each torus is presented as a lift of a contractible curve in a two-dimensional symplectic reduction; by choosing the curve to enclose area just $a$, the lift is squeezed into a narrow polydisk (or cylinder), whose displacement energy is known exactly. For the lower bound on $\hbar$, the paper uses a persistence lemma of Chekanov and Schlenk—holomorphic disks with boundary on a Lagrangian survive in sufficiently large Darboux charts—together with a positivity-of-intersections classification of the Maslov-2 disk classes, which for Brendel tori is carried out against carefully chosen complex hypersurfaces and yields $\hbar(\Upsilon_k(a))=\pi-ka$. The opposing semi-continuity properties of $e$ (upper) and $\hbar$ (lower) are the analytical tool that computes $\hbar$ in the non-monotone case and separates $e$ from $\hbar$ in the strict-inequality example.

What would settle it

Recompute the homology table in Proposition 4.2 with the apparent square-root typos in the disk maps for β1 and β2 corrected; if a Maslov-2 class with area strictly below π/(k+1) appears, then ℏ(Υ_k(π/(k+1)))<e(Υ_k(π/(k+1))), contradicting Theorem 1 for the monotone Brendel tori.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that Chekanov's inequality $\hbar(L)\le e(L)$ is not merely a formal bound: it is achieved exactly for two geometrically interesting families of exotic Lagrangian tori. For the small Chekanov tori in $\mathbb{C}P^n$, both invariants equal the area parameter $a$; for the monotone Brendel tori $\Upsilon_k(\pi/(k+1))$ in $\mathbb{C}^3$, both equal $\pi/(k+1)$. In the same Brendel family, the non-monotone members satisfy $\hbar(\Upsilon_k(a))=\pi-ka$, so the lower bound is known explicitly and is at most $a$, the upper bound on $e$ obtained by a squeezing argument. The paper further proves that no two Brendel tori with distinct parameters are symplectomorphic, and it identifies a Lagrangian $L_{\mathrm{FOOO}}=S^1(A)\times S^1_{\mathrm{eq}}\subset \mathbb{C}\times S^2(2a)$ for which $\hbar(L_{\mathrm{FOOO}})=a<e(L_{\mathrm{FOOO}})=A<\infty$, giving a positive answer to the question of whether Chekanov's bound can be strict for a displaceable Lagrangian.

Load-bearing premise

The Brendel-torus half of the main theorem rests on an unproved table (Proposition 4.2) listing which holomorphic disk classes bound those tori and what areas they have; the displayed formulas for two of the disk classes appear to contain typos, so a wrong table would break the equality e=ℏ for the monotone Brendel tori.

Editorial extensions

If this is right

For every small Chekanov torus $T^n_{\mathrm{PrCh}}(a)$ in $\mathbb{C}P^n$, the displacement energy is exactly $a$, so Chekanov's bound is sharp in arbitrarily high dimension.
For every monotone Brendel torus $\Upsilon_k(\pi/(k+1))$ in $\mathbb{C}^3$, the displacement energy is exactly $\pi/(k+1)$, extending the equality to an infinite family of exotic tori.
The formula $\hbar(\Upsilon_k(a))=\pi-ka$ gives an exact, computable lower bound for all non-monotone Brendel tori, leaving only the question of whether $e$ attains this lower bound.
The example $L_{\mathrm{FOOO}}$ shows that $\hbar<e<\infty$ can occur for a displaceable Lagrangian, so the question of whether a monotone example exists (Question 3) remains the next open step.
As a corollary, Brendel's tori $\Upsilon_k(a)$ are pairwise non-symplectomorphic across all parameters $(k,a)$, strengthening the known exotica.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same reduced-curve squeezing strategy should apply to other displaceable Lagrangian tori that arise as lifts from two-dimensional symplectic reductions, for example the displaceable Chekanov tori in $S^2\times S^2$; testing $e=\hbar$ there would be a direct extension.
If the conjectured disk-count correspondence with Auroux's tori holds, the monotone Brendel tori and Auroux's tori would be the same Lagrangians up to Hamiltonian isotopy, unifying two a priori different constructions of exotic tori in $\mathbb{C}^3$; the paper's count evidence supports but does not prove this.
The explicit window $\pi-ka\le e(\Upsilon_k(a))\le a$ suggests that non-monotone Brendel tori are natural candidates for additional examples of $\hbar<e$; if any such torus has $e>\pi-ka$, then strictness of Chekanov's bound is a phenomenon that persists away from the monotone endpoint.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the Chekanov-torus half of this paper is clean and complete, and the Brendel half is interesting but currently rests on an unproved and misprinted disk-class table. The conclusions are probably right, but the paper needs a real revision before I'd trust the Brendel results.

For small Chekanov tori in CP^n, Theorem 1 is solid. The upper bound e <= a comes from explicit Hamiltonian displacements of a suitably narrow lift; the lower bound hbar >= a uses Auroux's one-class computation and the Chekanov-Schlenk persistence lemma. Both steps are standard and carefully assembled. The same goes for Proposition 2.2 on lower semi-continuity of hbar; the proof via Fukaya's trick is terse but valid. And the clean computation hbar = a for the FOOO example is a genuinely useful add, since the literature only had e > hbar.

The problems are in Section 4. Proposition 4.2, which is load-bearing for everything about Brendel tori, is asserted as 'a straightforward computation' without proof. The displayed parametrizations are misprinted: the first coordinate should involve sqrt(1 - k|z|^2), not a/(1 - k|w0|^2). Taken literally, they are not Lagrangian for a general curve. The table of areas and Maslov indices is plausible once the coordinate is corrected, but the computation is not shown. Since Proposition 4.4, Corollary 4.5, and the classification all use this table, this is not a cosmetic issue.

The advertised new proof of exoticity for the monotone Brendel tori is explicitly 'modulo an argument for the regularity of the standard almost complex structure' (the author says so in the introduction). That means the paper does not actually deliver that proof as written. The evidence for the Auroux correspondence is a nice conjecture, but it is not a theorem.

I want to be clear: I did not find a contradiction, and the author is honest about the gaps. The Chekanov half is worth publishing on its own. The Brendel half needs the table verified and the regularity question addressed. I would send the paper to a careful referee, asking them to check Proposition 4.2 first. The right venue would be a symplectic geometry journal, and after revision it could be a solid contribution.

Referee Report

3 major / 4 minor

Summary. The paper studies two invariants of a closed Lagrangian submanifold: Hofer displacement energy e(L) and the minimal symplectic area hbar(L) of a pseudo-holomorphic disk (or sphere) with boundary on L. Chekanov's theorem gives hbar(L) <= e(L), and the paper computes both invariants for two families of Lagrangian tori: small Chekanov tori T_PrCh^n(a) in CP^n with 0<a<pi/(n+1), and Brendel tori Upsilon_k(a) in C^3 for k>=2 and pi/(k+1)<=a<pi/k. The main result, Theorem 1, states that e=hbar for the small Chekanov tori and for the monotone Brendel tori Upsilon_k(pi/(k+1)); for non-monotone Brendel tori the paper computes hbar(Upsilon_k)=pi-ka and leaves e as an open question. Section 2 establishes upper semi-continuity of e and a lower semi-continuity statement for hbar, and uses these to give a new computation of hbar for the FOOO example with e>hbar. The paper also derives constraints on the relative homology classes of holomorphic disks on Brendel tori, proposes a correspondence with Auroux's tori, and gives a new proof, conditional on a regularity argument, that the monotone Brendel tori are exotic.

Significance. If the results are correct, the paper provides some of the first computations showing equality e=hbar for non-standard, displaceable Lagrangian knots, and it gives a clean method for computing hbar via lower semi-continuity and explicit Hamiltonian displacements. The Chekanov-torus half of Theorem 1 appears structurally sound: the upper bound comes from an explicit displacement and the lower bound from Chekanov's inequality together with a Darboux-chart persistence lemma. The Brendel-torus half is plausible but rests on an unproved homology/Maslov table whose printed parametrizations contain evident typos. The paper is careful to rely on external benchmarks (Chekanov's inequality, product-torus results, Brendel's versal deformation computations) rather than fitting parameters, and it explicitly identifies its conditional claims. The significance of the exoticity claim is reduced by the fact that it is explicitly modulo regularity of the standard almost complex structure.

major comments (3)

[Sec. 4, Prop. 4.2] Proposition 4.2 is asserted with 'The proof is a straightforward computation', but it is the only verification that H_2(C^3, Upsilon_k) is spanned by alpha, beta1, beta2 with areas a, pi, 0 and Maslov indices 2, 2k+2, 0. This table is used in Corollary 4.3 to obtain e=hbar=pi/(k+1) for monotone Brendel tori and in Corollary 4.5 to obtain hbar(Upsilon_k)=pi-ka; without it, the Brendel half of Theorem 1 is unsupported. Moreover, the displayed parametrizations are not consistent with the level set nu_k^{-1}(pi,0) used in (9) and Appendix A: the first coordinate is written as a/(1-k|w|^2) (and similarly in beta1 and beta2), but the level-set relation r_1^2+k r_3^2=1 forces the first radius to be sqrt(1-k|w|^2), independent of a. As printed, the maps do not lie on Z_k and are not Lagrangian over a general curve Gamma. This is a verification gap rather than a demonstrated contradiction, but it must be closed by a complete computation or a precise reference before the Brendel results can be accepted.
[Abstract; Sec. 4.2.3] The abstract and the introductory summary advertise 'a new proof ... that Brendel's family of exotic tori consists of infinitely many distinct Lagrangians', but the argument in Section 4.2.3 is explicitly conditional: it assumes regularity of the standard almost complex structure J0 and that the existence of a holomorphic disk in one class beta1-k alpha+n0 beta2 persists when passing to the monotone limit. The text itself says 'modulo an argument for the regularity' and later uses the word 'Conjecturally'. The manuscript should state plainly in the abstract and introduction that this exoticity proof is conditional and not a theorem; as written, the claims exceed what is proved.
[Sec. 2, Prop. 2.2] The proof of Proposition 2.2 fixes an arbitrary tame almost complex structure J and chooses u_n with omega(u_n)=hbar_D(L_n,J), then asserts omega(v_n)->H. However, the hypothesis is convergence of hbar(L_n), which involves a supremum over J (and a minimum with the sphere area); for a fixed J the numbers hbar_D(L_n,J) need not converge to H. The proof should either choose J_n with hbar_D(L_n,J_n)->H and justify Gromov compactness for the pulled-back sequence of almost complex structures, or be rewritten via the 'Fukaya trick' in a way that avoids this step. In addition, the sentence 'Thus v is J-holomorphic, non-constant' is not justified if v is constant but one of the bubble disks g_k is nonconstant; the correct conclusion is that some nonconstant disk has area at most H. These are local, fixable gaps, but Proposition 2.2 is used in Example 2.3 and in the upper bound for hbar(Upsilon_k) in Corollary 4.5.

minor comments (4)

[Sec. 4.2.1] The expression 'epsilon P 8D' appears to be a typo for 'epsilon in partial D'; as printed it is unreadable.
[Cor. 4.5] The sentence 'A class of Maslov index 2 ell is of the form' contains a stray ell; the intended phrase is 'A class of Maslov index 2 is of the form'.
[Sec. 4.2.4] The table lists k=-1,0,1 even though the construction in Section 4 assumes k>=2; a sentence explaining the extended range and the meaning of Upsilon_{-1}, Upsilon_0, Upsilon_1 would help the reader.
[Sec. 4.2.2, Remark 4.7] The paragraph after Remark 4.7 states, without proof, that no two Upsilon_k(a) with distinct parameters are symplectomorphic even in the more general two-parameter notation; since this is a stronger classification statement, the parameter domain and the argument should be spelled out.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; e = hbar is obtained by matching an external lower bound (Chekanov) with an explicit displacement upper bound, and no fitted parameter is relabeled as a prediction.

full rationale

Walking the derivation chain, the two halves of Theorem 1 are built from independent external inputs. For small Chekanov tori in CP^n, the upper bound e <= a is an explicit displacement construction (Proposition 3.2), and the lower bound hbar >= a uses Chekanov–Schlenk persistence (Proposition 3.3) together with the known disk computation for Chekanov tori in C^n from Eliashberg–Polterovich and Auroux (Proposition 3.1). For monotone Brendel tori, Proposition 4.1 gives e <= a by making the lifted curve narrow, while hbar >= pi/(k+1) comes from the direct homology/Maslov table in Proposition 4.2 and monotonicity. Nothing is fitted to make the equality hold: the target identity follows because an independently computed minimal disk area coincides with a displacement-energy upper bound. The paper does rely on Brendel's published versal-deformation theorem [Bre25, Prop. 4.4] for the non-monotone upper bound hbar(Y_k) <= pi - k a, but that is a cited external result, and Brendel is an acknowledged collaborator rather than an author of this paper; this is legitimate use of prior work, not a self-citation chain that defines the conclusion. The one genuine weakness is Proposition 4.2: its homology/Maslov table is asserted as 'a straightforward computation' with no derivation, and the displayed representatives as printed contain a typo (the first coordinate should presumably involve sqrt(1 - k|z|^2), consistent with Appendix A). Corollaries 4.3 and 4.5 therefore inherit a verification gap. That is an omitted proof and a possible typo, however, not circularity: the table is a computation from the definitions of alpha, beta1, beta2, not an assumption of the theorem being proved. The FOOO example (Theorem 2) also uses an external lower bound for displacement energy and a Gromov-compactness argument, with no fitted input. Overall, the manuscript is self-contained against external benchmarks and exhibits no circular reasoning.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard symplectic geometry theorems (Chekanov's inequality, Gromov compactness) and on prior results about Chekanov and Brendel tori. No new entities or free parameters are introduced; the parameter a is a geometric input, not fitted.

assumptions (5)

standard math Chekanov's inequality hbar(L) <= e(L) for geometrically bounded symplectic manifolds
Used throughout as the lower bound on e; cited to [Che98].
domain assumption The Chekanov torus in C^n has exactly one Maslov-2 disk class with area a (Prop 3.1)
Cited to [EP97] and [Aur09]; provides the lower bound hbar_D(T_Ch^n(a), J0)=a in Prop 3.4.
domain assumption Persistence of hbar_D under Darboux embeddings (Prop 3.3, [CS16, Lemma 2.2])
Used to transfer disk-area lower bounds from C^n to CP^n in Prop 3.4.
domain assumption Brendel's construction of Upsilon_k and the versal deformation identifying L_t with product tori ([Bre25, Prop 4.4])
Used in Cor 4.5 to bound hbar(Upsilon_k) from above via lower semi-continuity.
standard math Gromov compactness and Fukaya's trick in Prop 2.2
Supports the lower semi-continuity of hbar; the trick is cited to [STV24].

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Cite this review

Pith. "Pith review of Tightness of Chekanov's bound on displacement energy for some Lagrangian knots." pith.science (2026). https://pith.science/paper/ZKLHOE4X

@misc{pith2026250721282,
  author       = {Pith},
  title        = {Pith review of: Tightness of Chekanov's bound on displacement energy for some Lagrangian knots},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZKLHOE4X}},
  note         = {Machine review of arXiv:2507.21282}
}

abstract

By a classical theorem of Chekanov, the displacement energy, $e$, of a Lagrangian submanifold is bounded from below by the minimal area of pseudo-holomorphic disks with boundary on the Lagrangian, $\hbar$. We compute $e$ and $\hbar$ for displaceable Chekanov tori in $\mathbb{C}P^n$, and for an infinite family of exotic tori in $\mathbb{C}^3$ constructed by Brendel. In these families, $e=\hbar$. We compare continuity properties of $e$ and $\hbar$ on the space of Lagrangians. This provides an example (suggested by Fukaya, Oh, Ohta, and Ono) where $e>\hbar$. Our calculations have further applications such as a new proof, inspired by work of Auroux, that Brendel's family of exotic tori consists of infinitely many distinct Lagrangians.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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Infinitely many monotone Lagrangian tori in R6

isbn: 978-1-57146-138-4. doi: 10.4310/SDG.2008.v13.n1.a1. [Aur14] Denis Auroux. “Infinitely many monotone Lagrangian tori in R6”. In: In- ventiones mathematicae 201.3 (Nov. 2014), pp. 909–924. issn: 1432-1297. doi: 10.1007/s00222-014-0561-9 . [Bre22] Jo´ e Brendel. “An introduction to the Chekanov torus”. In: Proceedings of the G¨ okova Geometry-Topology ...

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issn: 1935-2565. [Aur09] Denis Auroux. “Special Lagrangian fibrations, wall-crossing, and mirror symmetry”. In: Surveys in differential geometry. Vol. XIII. Geometry, anal- ysis, and algebraic geometry: forty years of the Journal of Differential Ge- ometry. Vol. 13. Surv. Differ. Geom. Int. Press, Somerville, MA, 2009, pp. 1–

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Holomorphic discs, spin structures, and Floer cohomol- ogy of the Clifford torus

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