Pith. sign in

REVIEW 2 major objections 5 minor 33 references

Hamiltonian linking and Symplectic packing

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A packing inequality or a Hamiltonian link: the dichotomy inside a symplectic ball.

desk verdict A short, honest note whose new packing-versus-linking dichotomy is elegant and likely correct, but the referee should check the exact hypotheses of the cited max formula and the spectral-diameter result. read the letter →

arxiv 2507.01416 v1 pith:QMXRL6KQ submitted 2025-07-02 math.SG

Dylan Cant This is my paper · ORCID
classification math.SG MSC 53D4053D3553D12
keywords HamiltonianlinkingsymplecticpackingspectralcapacitydiametermaximumformulaboundaryminimalityFloerhomologyJ-holomorphicdisks
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 establishes a dichotomy for two disjoint compact sets inside a standard symplectic ball: if the pair cannot be separated by a Hamiltonian diffeomorphism so that they lie on opposite sides of a hyperplane, then their spectral capacities must sum to at most the ball's capacity. Equivalently, any pair whose spectral capacities sum to more than the ball's capacity is necessarily Hamiltonian linked. The observation matters because it turns a packing obstruction into a linking statement and vice versa, giving a single inequality that controls both phenomena. From it the paper derives boundary minimality of the ball, a closing lemma for Reeb orbits, and bounds on areas of J-holomorphic disks with Lagrangian boundary.

What carries the argument

The machinery is spectral capacity $c(K)$, the Floer-theoretic invariant that assigns a nonnegative size to a compact set, together with the spectral diameter $\gamma(\Omega)$ of the space of isotopies supported in $\Omega$. The load-bearing identity is the maximum formula: for isotopies supported on opposite sides of a hyperplane, the spectral invariant of their product is the maximum of the two individual invariants. This converts the conjugated pair of isotopies into two product isotopies whose spectral invariants dominate the original capacities, and the spectral-diameter equality $\gamma(B(a))=a$ then bounds their sum. The proof of Theorem 1 reduces the whole packing-versus-linking alternative to exactly this formula together with the ball's spectral-diameter computation.

What would settle it

Find two disjoint compact subsets $K_1,K_2$ of $B(a)$ that can be separated by a Hamiltonian diffeomorphism to opposite sides of a hyperplane but satisfy $c(K_1)+c(K_2)>a$; Theorem 1 asserts no such pair exists.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: for compact disjoint $K_1,K_2\subset B(a)$ in the standard symplectic ball of capacity $a$, either $K_1$ and $K_2$ are Hamiltonian linked, or $c(K_1)+c(K_2)\le a$. Here Hamiltonian linked means the pair cannot be mapped by a compactly supported Hamiltonian diffeomorphism to opposite sides of a hyperplane. The proof conjugates by the diffeomorphism that separates the sets, applies the maximum formula to the conjugated isotopies, and uses the fact that the spectral diameter of the ball equals its capacity to bound the sum by $a$. A corollary is that any domain $\Omega$ with $c(\Omega)=\gamma(\Omega)$ is boundary minimal, so removing an open set touching the boundary strictly lowers its capacity.

Load-bearing premise

The argument depends on the maximum formula holding exactly: when two Hamiltonian isotopies have supports on opposite sides of a hyperplane, the spectral invariant of their product must equal the larger of the two individual invariants, not merely lie close to it or be bounded by it.

Editorial extensions

If this is right

  • A compact set $K\subset\partial B(a)$ with $c(K)=b$ is Hamiltonian linked with every smaller ball $B(c)\subset B(a)$ once $c>a-b$, giving a higher-dimensional camel-type obstruction (Corollary 2).
  • Any domain $\Omega$ with $c(\Omega)=\gamma(\Omega)$ is boundary minimal: deleting any open set that touches $\partial\Omega$ strictly lowers the capacity (Corollary 3).
  • Boundary minimality yields a closing-type lemma: for any nonempty open subset of the boundary there is a Reeb orbit through it whose period plus a cohomological correction is bounded by $c(\Omega)$ (Lemma 4).
  • For any Lagrangian in the interior of $B(a)$ and any admissible almost complex structure, either the Lagrangian bounds a $J$-holomorphic disk of area at most $a-b$, or it is linked with a fixed boundary set $K$ of capacity $b$ (Corollary 6).
  • A uniformly convex domain is Zoll if and only if it is boundary minimal (Theorem 5).

Reading between the lines

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

  • If the maximum formula extends to a wider class of disjoint pairs, the same proof would give the packing inequality $c(K_1)+c(K_2)\le a$ for those pairs; the paper explicitly notes exact and boundary-$\pi_1$-injective Liouville embeddings as known cases.
  • An affirmative answer to the paper's Question 1 would settle the conjecture that every Lagrangian in the interior of $B(a)$ bounds $J$-holomorphic disks of area at most $a/2$, since a boundary set of capacity at least $a/2$ would force the alternative in Corollary 6 to fail.
  • Theorem 13 shows that sets of arbitrarily small positive spectral capacity can link the ball, so any future criterion for linking must use more than positivity of capacity; this sharpens the open question of whether zero-capacity sets can link.
  • The same dichotomy may be testable in low dimensions by computing spectral capacities of explicit toric subsets of the ball, since Theorem 9 already computes capacities of boundary subsets cut out by toric inequalities.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper defines two compact sets in R^{2n} to be Hamiltonian unlinked if some compactly supported Hamiltonian diffeomorphism sends them to opposite sides of a hyperplane, and Hamiltonian linked otherwise. Its main result (Theorem 1) states that if two disjoint compact subsets of the standard ball B(a) are Hamiltonian unlinked, then the sum of their spectral capacities is at most a. The proof conjugates by an unlinking diffeomorphism, applies a maximum formula for spectral invariants of disjointly supported isotopies and the spectral-diameter bound for the ball, and concludes the capacity inequality. The paper then derives several corollaries, including boundary minimality of the ball, a closing lemma that leads to a characterization of uniformly convex Zoll domains, a statement about areas of J-holomorphic disks bounded by Lagrangians, and two further results (Theorems 9 and 12) about capacities of toric subsets of the boundary and about Reeb chords between unlinked Legendrians.

Significance. If Theorem 1 is established, it is a clean and potentially useful observation connecting symplectic packing inequalities with Hamiltonian linking. The proof is remarkably short and the applications are wide-ranging. The main obstruction to accepting the paper as it stands is that the central maximum-formula lemma is not proved, and the cited literature does not clearly support the exact equality used. The paper also depends essentially on the spectral-diameter theorem of the author's own [AAC24]. The manuscript is clearly written and the main idea is elegant, but the central claim requires a missing verification.

major comments (2)
  1. [§2.2 (Lemma 15)] Lemma 15 is the only input that converts the conjugated isotopies into quantities controlled by the spectral diameter. In the proof of Theorem 1 (§2.4), the equality c(φ0φ1) = max{c(φ0), c(φ1)} is applied to isotopies supported on opposite sides of a hyperplane, and only the lower bounds c(φ1φ2^{-1}) ≥ c(φ1) and c(φ2φ1^{-1}) ≥ c(φ2) are actually needed. The lemma is stated without proof and with a citation to [HLRS16, GT23]. The discussion in §1.1 suggests those references prove a max inequality for generating-function capacities of subsets, or for disjointly supported Hamiltonians under additional hypotheses, and the author explicitly notes in §1.3.5 that a max inequality is insufficient for the argument. The exact equality for spectral invariants of arbitrary compactly supported isotopies with supports on opposite sides of a hyperplane, in the Floer convention of [AAC24, CZ24], needs a complete proof or a precise theorem-level reference with all hypotheses verified. This is load-bearing: without this equality, Theorem 1 does not follow.
  2. [§2.3 and §2.4 (Lemma 16 and its use)] Lemma 16 is stated for isotopies supported in B(a), but the proof of Theorem 1 is written for a domain Ω with a = c(Ω) = γ(Ω), and the final bound c(φ1φ2^{-1}) + c(φ2φ1^{-1}) ≤ a is applied to an isotopy supported in U1 ∪ U2 ⊂ Ω. Please state the version needed for arbitrary Ω with γ(Ω) = a, or restrict the proof (and statement) to Ω = B(a). Since Lemma 16 is cited from the author's own preprint [AAC24], the manuscript should also make the precise dependence clear, for example by giving a theorem statement and a reference with a theorem number.
minor comments (5)
  1. [§2.4] The passage from disjoint compact sets K1, K2 to disjoint open neighborhoods U1, U2 that are 'also unlinked' is true but should be justified explicitly by applying the unlinking diffeomorphism and taking preimages of separating neighborhoods.
  2. [§2.8 (Theorem 9)] The notation E(...) is not defined, and the identities a = c(K ∪ K^c) and c(K ∪ K^c) ≤ c(K) + c(K^c) are used without proof; these steps need justification, as they are not standard consequences of spectral capacity alone.
  3. [§2.9 (Theorem 12)] The construction from [Moh01] is not described in enough detail to verify that the two Lagrangians are disjoint, unlinked, and have capacities arbitrarily close to the minimal Reeb chord length; please expand this argument.
  4. [§1.1 (Corollary 3)] The assertion that a small ball inside U ∩ Ω is unlinked with Ω \ U is not immediate and should be proved or supported by a reference.
  5. [Throughout] There are minor typographical issues, including 'proves the the maximum formula' in §1.1 and 'pertubations' in §1.3.3.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: Theorem 1 is a genuine deduction from the cited maximum formula and the spectral-diameter/capacity equality; the load-bearing self-citation to [AAC24] is an independent external result, not a repackaging of the conclusion.

full rationale

The main proof (§2.4) assumes K1,K2 are Hamiltonian unlinked, chooses disjoint unlinked open neighborhoods U1,U2, and bounds c(K1)+c(K2) by sup c(φ1)+sup c(φ2) over isotopies supported in U1,U2. It then applies Hamiltonian invariance (Lemma 14) and the maximum formula (Lemma 15) to obtain c(φ1)+c(φ2) ≤ c(φ1φ2^{-1})+c(φ2φ1^{-1}), and uses γ(Ω)=c(Ω)=a (Lemma 16, quoted from [AAC24]) for the final bound. Each of these inputs is an external theorem with stated hypotheses; no parameter is fitted, no definition is chosen so that the packing inequality is true by construction, and no quantity announced as a prediction is derived from the same data. The only self-citation in the derivation is Lemma 16, which is load-bearing but independent: it is a parameter-free statement about the spectral diameter of the ball proved in a separate paper ([AAC24]); this note does not assume Theorem 1 in invoking it, so the citation is not a circular reduction under the hard rules. The paper itself flags the main fragility: Lemma 15 is asserted by citation ("this result is not new (and is proved in [HLRS16, GT23])"), and §1.3.5 explicitly notes that the max inequality alone would not suffice and the equality is required. If the cited equality fails in the paper's Floer-theoretic convention, Theorem 1 would not follow; that is a correctness risk, not an exhibitable circular step. The corollaries are contrapositives or direct applications of Theorem 1, and Theorem 9 uses an additional subadditivity/commuting-type inequality rather than assuming the packing inequality. Therefore no circular step is identified.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted numbers appear: the parameters a, b1, b2, epsilon, and c are theorem variables, not free constants. The main assumptions are black-box theorems from prior literature, listed above. One of these, the spectral diameter result, comes from a paper co-authored by the present author, so it is a self-citation; it is treated as independent because it is established in a separate paper. No new entities are postulated, and 'Hamiltonian linked' is a definition, not an entity with independent evidence requirements.

assumptions (6)
  • domain assumption Maximum formula for spectral invariants: for compactly supported isotopies with supports on opposite sides of a hyperplane, c(phi0 phi1) = max(c(phi0), c(phi1)).
    Cited to [HLRS16, GT23, Tan22]; the proof of Theorem 1 reduces to this equality. It is not proved in the note.
  • domain assumption Spectral diameter equals capacity for the ball: gamma(B(a)) = c(B(a)) = a, so for every isotopy phi supported in B(a), c(phi) + c(phi^{-1}) <= a.
    Cited to [AAC24], co-authored by the present author; it provides the final bound in the proof of Theorem 1 and is not reproved here.
  • standard math Spectral capacity is invariant under conjugation by Hamiltonian diffeomorphisms.
    Treated as standard and proved by a continuity and action-spectrum argument in Lemma 14, citing [Vit92, Corollary 4.3].
  • domain assumption For a Lagrangian L, its spectral capacity controls the area of J-holomorphic disks it bounds, via [Her04, CZ24].
    Used in Corollary 6 and Corollary 11 to translate capacity bounds into disk-area bounds.
  • domain assumption For a Liouville domain, the capacity of the subdomain Omega(sf) equals the period of a Reeb orbit for e^{-sf} alpha plus Gamma(gamma_s), and s maps to c(Omega(sf)) is continuous.
    Quoted as well-known in the proof of Lemma 4 and needed for the closing lemma.
  • domain assumption For uniformly convex domains, c(Omega) equals the shortest period of a Reeb orbit [AK22], and uniformly convex Zoll domains are local systolic maximizers [AB23].
    This couples Lemma 4 to the Zoll characterization in Theorem 5.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Hamiltonian linking and Symplectic packing." pith.science (2026). https://pith.science/paper/QMXRL6KQ

@misc{pith2026250701416,
  author       = {Pith},
  title        = {Pith review of: Hamiltonian linking and Symplectic packing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QMXRL6KQ}},
  note         = {Machine review of arXiv:2507.01416}
}
read the original abstract

In this short note, we make an observation relating symplectic packings of the standard symplectic ball by two sets and Hamiltonian linking.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 29 canonical work pages

  1. [1]

    The spectral diameter of a symplectic ellipsoid

    H. Alizadeh, M. S. Atallah, and D. Cant, The spectral diameter of a symplectic ellipsoid, arXiv:2408.07214, August 2024

    work page Pith review arXiv 2024

  2. [2]

    Abbondandolo and G

    A. Abbondandolo and G. Benedetti, On the local systolic optimality of Z oll contact forms , Geom. Funct. Anal. 33 (2023), no. 2, 299--363, doi:10.1007/s00039-023-00624-z https://doi.org/10.1007/s00039-023-00624-z. 4578460

    work page doi:10.1007/s00039-023-00624-z 2023

  3. [3]

    Abbondandolo and J

    A. Abbondandolo and J. Kang, Symplectic homology of convex domains and C larke's duality , Duke Math. J. 171 (2022), no. 3, 739--830, doi:10.1215/00127094-2021-0025 https://doi.org/10.1215/00127094-2021-0025. 4382979

    work page doi:10.1215/00127094-2021-0025 2022

  4. [4]

    Audin, Fibrés normaux d'immersions en dimension double, point doubles d'immersions lagrangiennes et plongements totalement réels, Comment

    M. Audin, Fibrés normaux d'immersions en dimension double, point doubles d'immersions lagrangiennes et plongements totalement réels, Comment. Math. Helv. 63 (1988), 593--623

    work page 1988

  5. [5]

    Buhovsky and S

    L. Buhovsky and S. Tanny, A local-to-global inequality for spectral invariants and an energy dichotomy for F loer trajectories , J. Fixed Point Theory Appl. 27 (2025), no. 3, 1--37, doi:10.1007/s11784-024-01153-3 https://doi.org/10.1007/s11784-024-01153-3. 4844096

    work page doi:10.1007/s11784-024-01153-3 2025

  6. [6]

    Chaidez, I

    J. Chaidez, I. Datta, R. Prasad, and S. Tanny, Contact homology and higher dimensional closing lemmas, J. Mod. Dyn. 20 (2024), 67--153, doi:10.3934/jmd.2024003 https://doi.org/10.3934/jmd.2024003

    work page doi:10.3934/jmd.2024003 2024

  7. [7]

    Cieliebak and K

    K. Cieliebak and K. Mohnke, Punctured holomorphic curves and L agrangian embeddings , Invent. math. 212 (2018), 213--295

    work page 2018

  8. [8]

    On the spectral capacity of submanifolds

    D. Cant and J. Zhang, On the spectral capacity of submanifolds, arXiv:2409.14142, September 2024, Submitted

    work page Pith review arXiv 2024

Show all 33 references
  1. [9]

    Dimitroglou Rizell, Uniqueness of extremal L agrangian tori in the four-dimensional disc , Proceedings of the G \"o kova Geometry-Topology Conference 22 (2015), 151--167

    G. Dimitroglou Rizell, Uniqueness of extremal L agrangian tori in the four-dimensional disc , Proceedings of the G \"o kova Geometry-Topology Conference 22 (2015), 151--167

    2015

  2. [10]

    Faisal, Extremal L agrangian tori in toric domains , arXiv:2504.13076, 04 2025, doi:10.48550/arXiv.2504.13076 https://doi.org/10.48550/arXiv.2504.13076

    S. Faisal, Extremal L agrangian tori in toric domains , arXiv:2504.13076, 04 2025, doi:10.48550/arXiv.2504.13076 https://doi.org/10.48550/arXiv.2504.13076

    2025 doi arXiv

  3. [11]

    V. L. Ginzburg, B. Z. G\"urel, and M. Mazzucchelli, On the spectral characterization of B esse and Z oll R eeb flows , Ann. Inst. H. Poincar\'e C Anal. Non Lin\'eaire 38 (2021), no. 3, 549--576, doi:10.1016/j.anihpc.2020.08.004 https://doi.org/10.1016/j.anihpc.2020.08.004. 4227045

    2021 doi

  4. [12]

    Gutt and M

    J. Gutt and M. Hutchings, Symplectic capacities from positive S^ 1 -equivariant symplectic homology , Alg. Geom. Topol. 18 (2018), 3537--3600

    2018

  5. [13]

    V. L. Ginzburg, Coisotropic intersections, Duke Math. J. 140 (2007), no. 1, 111--163

    2007

  6. [14]

    Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent

    M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307--347

    1985

  7. [15]

    Ganor and S

    Y. Ganor and S. Tanny, Floer theory of disjointly supported Hamiltonians on symplectically aspherical manifolds , Alg. Geom. Topol. 23 (2023), no. 2, 645--732

    2023

  8. [16]

    Hermann, Inner and outer H amiltonian capacities , Bull

    D. Hermann, Inner and outer H amiltonian capacities , Bull. Soc. math. France 132 (2004), no. 4, 509--541

    2004

  9. [17]

    Humili \`e re, F

    V. Humili \`e re, F. Le Roux, and S. Seyfaddini, Towards a dynamical interpretation of H amiltonian spectral invariants on surfaces , Geom. Topol. 20 (2016), no. 4, 2253--2334

    2016

  10. [18]

    Irie, Dense existence of periodic R eeb orbits and ECH spectral invariants , J

    K. Irie, Dense existence of periodic R eeb orbits and ECH spectral invariants , J. Mod. Dyn. 9 (2015), 357--363, doi:10.3934/jmd.2015.9.357 https://doi.org/10.3934/jmd.2015.9.357

    2015 doi

  11. [19]

    Kang, On the strong Arnold chord conjecture for convex contact forms , arXiv:2304.07016, 2023, doi:10.48550/arXiv.2304.07016 https://doi.org/10.48550/arXiv.2304.07016

    J. Kang, On the strong Arnold chord conjecture for convex contact forms , arXiv:2304.07016, 2023, doi:10.48550/arXiv.2304.07016 https://doi.org/10.48550/arXiv.2304.07016

    2023 doi

  12. [20]

    Kang and J

    J. Kang and J. Zhang, Strong A rnold chord conjecture via normalized capacities , arXiv:2404.05150, 2024, doi:10.48550/arXiv.2404.05150 https://doi.org/10.48550/arXiv.2404.05150

    2024 doi arXiv

  13. [21]

    Laudenbach and J.-C

    F. Laudenbach and J.-C. Sikorav, Hamiltonian disjunction and limits of L agrangian submanifolds , Int. Math. Res. Not. 1994 (1994), no. 4, 161--168

    1994

  14. [22]

    S. Matijevi\'c, Positive ( S^ 1 -equivariant) symplectic homology of convex domains, higher capacities, and C larke's duality , arXiv:2410.13673v2, 10 2024, doi:10.48550/arXiv.2410.13673 https://doi.org/10.48550/arXiv.2410.13673

    2024 doi arXiv

  15. [23]

    , Systolic S^ 1 -index and characterization of non-smooth Z oll convex bodies , arXiv:2501.13856v2, 2 2025, doi:10.48550/arXiv.2501.13856 https://doi.org/10.48550/arXiv.2501.13856

    2025 doi

  16. [24]

    Mohnke, Holomorphic disks and the chord conjecture, Ann

    K. Mohnke, Holomorphic disks and the chord conjecture, Ann. Math. 154 (2001), 219--222, doi:10.2307/3062116 https://doi.org/10.2307/3062116

    2001 doi

  17. [25]

    McDuff and D

    D. McDuff and D. Salamon, Introduction to symplectic topology, 3rd ed., Oxford Graduate Texts in Mathematics, no. 27, Oxford University Press, 2012

    2012

  18. [26]

    Polterovich, Monotone L agrange submanifolds of linear spaces and the M aslov class in cotangent bundles , Math

    L. Polterovich, Monotone L agrange submanifolds of linear spaces and the M aslov class in cotangent bundles , Math. Z. 207 (1991), no. 2, 217--222

    1991

  19. [27]

    Tanny, A max inequality for spectral invariants of disjointly supported H amiltonians , J

    S. Tanny, A max inequality for spectral invariants of disjointly supported H amiltonians , J. Symp. Geom. 20 (2022), no. 5, 1159--1213

    2022

  20. [28]

    Viterbo, Plongements lagrangiens et capacités symplectiques de tores dans ^ 2n , C

    C. Viterbo, Plongements lagrangiens et capacités symplectiques de tores dans ^ 2n , C. R. Acad. Sci. Paris 311 (1990), 487--490

    1990

  21. [29]

    , Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992), no. 1, 685--710

    1992

  22. [30]

    , Functors and computations in F loer homology with applications, part II , arXiv:1805.01316v1, 1996

    1996 arXiv

  23. [31]

    Weinstein, Lectures on symplectic manifolds, Regional Conference Series in Mathematics, vol

    A. Weinstein, Lectures on symplectic manifolds, Regional Conference Series in Mathematics, vol. 29, American Mathematical Society, Providence, RI, 1977

    1977

  24. [32]

    Xue, Closing lemma and KAM normal form , arXiv:2207.06208, 2022, doi:10.48550/arXiv.2207.06208 https://doi.org/10.48550/arXiv.2207.06208

    J. Xue, Closing lemma and KAM normal form , arXiv:2207.06208, 2022, doi:10.48550/arXiv.2207.06208 https://doi.org/10.48550/arXiv.2207.06208

    2022 doi arXiv

  25. [33]

    Ziltener, On the strict Arnold chord property and coisotropic submanifolds of complex projective space , Int

    F. Ziltener, On the strict Arnold chord property and coisotropic submanifolds of complex projective space , Int. Math. Res. Not. 2016 (2016), no. 3, 795--826, doi:10.1093/imrn/rnv153 https://doi.org/10.1093/imrn/rnv153

    2016 doi

Pith tools

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