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On the sharp constants in curl-Sobolev inequalities on $\mathbb{S}^n$

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Pith's one-line read On the n-sphere with n ≡ 3 mod 4, the paper proves that Killing (n−1)/2-forms are strict local minimizers of the curl-Sobolev quotient J1, while for the quotient J2 they are unstable, making the sharp constant for J2 strictly smaller than t

desk verdict Solid spectral analysis and a likely-correct instability result, but Theorem 1.1's proof is delegated to an unpublished appendix; needs revision before it is complete. read the letter →

arxiv 2607.19091 v1 pith:GQDLRHAJ submitted 2026-07-21 math.DG math.APmath.SP

Guofang Wang , Mingwei Zhang This is my paper · ORCID
classification math.DGmath.APmath.SP MSC 35A2346E3558A10
keywords curloperatorKillingformsconformallyinvariantSobolevinequalitiessharpconstantsstabilitydifferentialsphereminimizers
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

The paper studies conformally invariant Sobolev inequalities for the middle-degree curl operator on the standard sphere. It tries to establish that the natural candidate extremals—Killing forms and their conformal images—are locally optimal for one such inequality (J1) but not for another (J2). If correct, this gives a quantitative local stability estimate for J1 and disproves a conjectured formula for the sharp constant of J2. The same instability extends to a broader family of form-valued Sobolev quotients. For dimension 3, the paper also improves the known lower bound for the J1 sharp constant by relating it to a shifted spinorial inequality.

What carries the argument

The central object is the middle-degree curl operator curl = ∗d acting on (n−1)/2-forms, together with the two conformally invariant quotients J1 and J2. The argument relies on the conformal invariance of these quotients, the Hodge decomposition (so that exact forms can be ignored), and a classification of curl eigenspaces on the sphere. The key quantitative input is a set of sharp L2 estimates—especially Proposition 3.4—that control the component of an eigenform along a fixed Killing form; these estimates yield a spectral gap for the second variation of J1 in directions transverse to the conformal family M. This spectral gap is then upgraded to a nonlinear stability estimate.

What would settle it

Use the second-variation formula in Proposition 4.3 with a direction φ ∈ E2 ∩ Q^⊥: if one can find such a φ for which the quadratic form G(φ) is negative (or equality cases in Proposition 3.4(2) that are not contained in Q), then the spectral gap in Theorem 4.4 fails and Theorem 1.1 collapses. Conversely, a direct check that G(φ) remains positive for all φ in that subspace for some n ≡ 3 mod 4 would support the claim.

Watch

Extended reading notes

Core claim

For n ≡ 3 mod 4, every Killing (n−1)/2-form and its conformal images form a family M of critical points for both J1 and J2, analogous to the classical extremal family of the scalar Sobolev inequality. The paper proves that near M, J1 satisfies a quantitative local stability estimate: if a form is close to M in the curl-norm, then the excess of J1 over its value at Killing forms is bounded below by a positive constant times the distance to M, squared. In particular, every element of M is a strict local minimizer of J1 in the conformally invariant space W^{1,2n/(n+1)}. For J2, the paper shows the opposite: each such critical point is unstable, and there exist variations that strictly decrease

Load-bearing premise

The proof of Theorem 1.1 ends by invoking, without reproducing, a 'standard quantitative argument' from the authors' companion work to pass from a spectral gap of the second variation to the nonlinear stability estimate; if that argument does not carry over verbatim—especially because curl has an infinite-dimensional kernel of exact forms—the quantitative stability and strict local minimality claimed in Theorem 1.1 are not established by the submitted manuscript.

Editorial extensions

If this is right

  • If Theorem 1.1 is correct, the Killing family M is locally the correct extremal family for J1, and a quantitative stability inequality holds in W^{1,2n/(n+1)}.
  • By conformal invariance, the local stability and instability results transfer from the sphere to R^n.
  • The instability for J2 shows that the sharp constant S2 is strictly less than (n+1)^2/4 ω_n^{2/n}, disproving the natural conjecture that the Killing forms attain it.
  • The instability extends to the generalized quotients J_{p,k} for each Killing (n−1)/2-form and for all 1 < p < n, as well as to the 1-form quotient with p = n/2.
  • A footnote reports that the authors have since confirmed global minimality of M for J1 in a later preprint; if that holds, the sharp constant S1 would equal (n+1)/2 ω_n^{1/n} and the quantitative stability would be global.

Reading between the lines

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

  • The different behavior of J1 and J2 highlights the role of the gauge optimization in the denominator: for J1, the denominator is the helicity-type integral ⟨curl α, α⟩, while for J2 it is an infimum over exact perturbations; the latter creates negative directions that destroy local minimality.
  • The expected Morse index of each Killing form for J2 (the paper suggests n+1) could be computed explicitly; if confirmed, it would quantify the instability and could guide the search for true minimizers.
  • Since the paper leaves open whether the infimum S2 is attained, a natural next question is whether minimizers of J2 exist and are non-invariant under the symmetry group that fixes the Killing forms.
  • The Appendix C lower bound for n = 3, obtained via a shifted Dirac operator, could be sharpened further if the spinorial comparison in Lemma C.2 can be made into an equality case; that would pinpoint the corrections to the conjectured value of S1.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 6 minor

Summary. The paper studies two conformally invariant Sobolev quotients for middle-degree forms on S^n, n≡3 mod 4: the curl–helicity quotient J1 and the gauge-optimized quotient J2. The main positive claim (Theorem 1.1) is a quantitative local stability estimate for J1 around the conformal family M of Killing (n−1)/2-forms and their conformal images, implying strict local minimality in W^{1,2n/(n+1)}. The paper also claims instability of this same family for J2 and for a family of generalized quotients J_{p,k}, thereby giving a strict upper bound for the sharp constant S2 and disproving the conjectured equality (1.4). Supporting material includes sharp L2 estimates for eigenforms, a classification of curl eigenforms, a conformal invariant related to the first curl eigenvalue, and a lower bound for S1 in dimension 3 by comparison with the Dirac operator. The instability computations are mostly explicit, but the proof of the central quantitative stability theorem is not contained in the manuscript: it delegates the nonlinear step to an unpublished preprint.

Significance. If Theorem 1.1 were fully established, it would be a genuine analogue for the curl operator of the Aubin–Talenti stability theory, and the claimed local minimality of the conformal Killing family in the conformally invariant topology would be a substantial result. The paper contains several correct and useful ingredients: the second-variation formula, the L2 spectral gap in Theorem 4.4, the sharp L2 eigenform estimates in Proposition 3.4, and explicit negative directions for J2 and for the generalized quotients. The disproof of the conjecture S2 = ((n+1)^2/4)ω_n^{2/n} is also significant. However, the strongest theorem is currently conditional on an argument that is not reproduced, so the significance of the paper as a whole is not yet realized in the submitted version.

major comments (1)
  1. [Section 4, proof of Theorem 1.1] The nonlinear upgrade from the L2 spectral gap in Theorem 4.4 to the quantitative W^{1,2n/(n+1)} estimate in Theorem 1.1 is not proved in the manuscript. The text states that a 'standard quantitative argument (following [FZ22])' applies and that the implementation is 'identical to [WZ25, Appendix A]'. This is load-bearing, and it is not literally automatic: [WZ25] is an unpublished preprint treating the Dirac operator, whose kernel is trivial, whereas the curl operator has the infinite-dimensional kernel of exact forms; moreover the distance in Theorem 1.1 is measured through the L^{2n/(n+1)} curl norm normalized by the helicity denominator, not through the L2 curl norm used in Theorem 4.4. Footnote 1 further indicates that the global-minimizer and stability statement was obtained in a subsequent preprint. As written, the proof of Theorem 1.1 is incomplete; the nonlinear argument must be
minor comments (6)
  1. [Section 1, definition of J_{p,k}] The statement 'when k=(n−1)/2 and p=2n/(n+1), one has J_{2n/(n+1),(n−1)/2}=J2' is false. For these parameters q=2n/(n−1), so the numerator of J_{p,k} has exponent q/p=(n+1)/(n−1), while J2 has numerator exponent (n+1)/n; the denominators also differ, since J2 contains the gauge infimum and J_{p,k} does not. The J2 instability is proved separately in Section 5, so this does not destroy Theorem 1.2, but the claimed identification is a mathematical error and should be corrected.
  2. [Section 5, end] The sentence 'By conformal invariance, Proposition 5.1 implies Theorem 1.2' overstates the implication. Proposition 5.1 directly proves the J2 part of Theorem 1.2; the J_{p,(n−1)/2} part is proved later in Section 6 using different computations. The text should be reworded to avoid the suggestion that the full theorem follows from Proposition 5.1 alone.
  3. [Section 6.1, Lemma 6.1] The proof of Lemma 6.1 is only sketched, with several identities stated after 'direct (though somewhat lengthy) computations'. Since Theorem 1.3 rests on these identities, the authors should either provide the full verification or clearly mark the lemma as a computational assertion with details available in a supplement.
  4. [Section 1, footnote 1] The paragraph before the footnote says the authors 'cannot prove or disprove' global minimality of J1, while the footnote announces that the problem was solved in a subsequent preprint and that quantitative stability follows. This is confusing in a manuscript whose Theorem 1.1 already claims quantitative stability. Please clarify the logical status of Theorem 1.1 with respect to the later preprint.
  5. [Section 5, Proposition 5.1] For the record, the denominator estimate (∫|α−dφ|^{2n/(n−1)})^{(n−1)/n} ≥ ω_n^{−1/n}∫|α−dφ|^2 is correct by Hölder; the equality analysis in Proposition 5.1 is also consistent. I do not find the error in this step that a preliminary review note suggested.
  6. [Theorem 1.1 statement] There is a typo: 'for any α∈W^{1,2n/(n+1)} with with ∫<curlα,α>>0' should read 'with' once. Similar small typographical issues occur elsewhere (e.g., 'particulary').

Circularity Check

1 steps flagged · score 4.0 of 10

Theorem 1.1's nonlinear upgrade is delegated to the authors' own [WZ25] appendix; the L^2 spectral gap is proved, but the central W^{1,2n/(n+1)} stability estimate is not derived in this manuscript.

  1. self citation load bearing [Section 4, Proof of Theorem 1.1 (after Theorem 4.4)]
    "A standard quantitative argument (following [FZ22]) then upgrades this spectral gap to the nonlinear estimate in Theorem 1.1; in our setting the implementation is identical to [WZ25, Appendix A]."

    The quantitative stability estimate in Theorem 1.1 is the paper's main positive result. Its proof does not contain the nonlinear argument; it is delegated to the authors' own unpublished preprint [WZ25, Appendix A]. The L^2 spectral gap proved here (Theorem 4.4) is a different object from the W^{1,2n/(n+1)} nonlinear estimate, and the needed quotient by the infinite-dimensional kernel ker(d) of exact forms is not shown to be covered by the asserted 'identical' transfer. Footnote 1 admits that global minimization/stability was only completed in a later preprint. Thus the central claim reduces to a self-citation that is not independently established in the manuscript.

full rationale

The paper contains substantial independent derivation: Proposition 3.4 and Theorem 4.4 are proved from Weitzenböck identities, eigenform classifications, and spectral decompositions; Proposition 5.1 uses Hölder and Hodge decomposition; Appendix C uses the external sharp spinorial Sobolev inequality [Amm03]. No parameter is fitted and no equation is imported that is equivalent to the target inequality. The J2 = J_{2n/(n+1),(n-1)/2} identification in Section 6 and the Hölder estimate in Proposition 5.1 are correctness concerns, not circularity. The score is raised above 2 because the nonlinear upgrade in Theorem 1.1 is load-bearing and rests on [WZ25], a self-citation whose applicability to the curl setting (with its infinite-dimensional kernel and different scaling) is asserted but not demonstrated in the submitted text.

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

The paper introduces no physical or geometric entities beyond the defined families M and Qξ, which are subsets of known objects. There are no fitted parameters: the constants are analytic candidates from a sharp-constant problem. The central claims rest on standard spectral geometry plus an external stability argument from the authors' own unpublished work, which is the main circularity burden.

assumptions (5)
  • standard math The spectrum of curl on (S^n,g_st) is {±((n+1)/2+k) : k≥0} with the multiplicities stated in Proposition 2.2
    Invoked throughout Sections 3-4 to decompose forms into eigenspaces E±k and to compute the dimension of E1; cited from [IK79] and [Bär19].
  • standard math The Weitzenböck formula and Hodge decomposition on the sphere (equations (2.1), (2.2))
    Used for eigenvalue computations, the identity curl²=Δ on co-closed forms, and the reduction modulo closed forms.
  • ad hoc to paper The nonlinear quantitative stability argument of [FZ22]/[WZ25, Appendix A] applies verbatim to the curl functional J1
    The proof of Theorem 1.1 after the spectral gap is not written; the text says the implementation is identical to the authors' unpublished preprint [WZ25, Appendix A].
  • domain assumption Sharp spinorial Sobolev inequality (C.1) from [Amm03]
    Used in Appendix C to obtain the lower bound S1 ≥ 3/2 ω_3^{1/3} in dimension 3.
  • domain assumption The conformal covariance formula for the Dirac operator and the stated spinor scaling used in Appendix C
    The construction of the minimizing sequence for the shifted Dirac quotient relies on a specific conformal transformation of spinors; the proof is only sketched.

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Pith. "Pith review of On the sharp constants in curl-Sobolev inequalities on $\mathbb{S}^n$." pith.science (2026). https://pith.science/paper/GQDLRHAJ

@misc{pith2026260719091,
  author       = {Pith},
  title        = {Pith review of: On the sharp constants in curl-Sobolev inequalities on $\mathbbS^n$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GQDLRHAJ}},
  note         = {Machine review of arXiv:2607.19091}
}
abstract

Let $n\equiv 3\ (\mathrm{mod}\ 4)$ and set $p=\frac{n-1}{2}$. On an oriented Riemannian $n$-manifold we consider the (middle-degree) curl operator, $\mathrm{curl}:*\mathrm{d}:\Omega^{p}\rightarrow\Omega^{p}$, and the associated conformally invariant Sobolev quotients on $(\mathbb{S}^n,g_{\mathrm{st}})$, \[ J_1(\alpha)=\frac{\big(\int|\mathrm{curl}\alpha|^{\frac{2n}{n+1}}\,\mathrm{dV}\big)^{\frac{n+1}{n}}}{\int\langle\mathrm{curl}\alpha,\alpha\rangle\,\mathrm{dV}}, \qquad J_2(\alpha)=\frac{\big(\int|\mathrm{curl}\alpha|^{\frac{2n}{n+1}}\,\mathrm{dV}\big)^{\frac{n+1}{n}}}{\inf_{\phi}\big(\int|\alpha-\mathrm{d}\phi|^{\frac{2n}{n-1}}\,\mathrm{dV}\big)^{\frac{n-1}{n}}}. \] Killing $p$-forms and their conformal images form a natural family of critical points for both functionals, analogous to the Aubin-Talenti family in the classical Sobolev inequality. We prove a quantitative local stability estimate for $J_1$ around this family, which in particular implies that every such form is a strict local minimizer in the conformally invariant space $W^{1,\frac{2n}{n+1}}$. In contrast, we show that these critical points are unstable for $J_2$ (and for related conformally invariant quotients), yielding a strict upper bound for the sharp constant of the $J_2$ inequality. By conformal invariance, the results on $\mathbb{S}^n$ transfer naturally to $\mathbb{R}^n$.

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  1. The sharp curl-Sobolev inequality

    math.DG 2026-07 conditional novelty 8.0 of 10

    For n ≡ 3 (mod 4), the sharp constant of the conformal curl–Sobolev quotient on S^n is (n+1)/2 ω_n^{1/n}, attained exactly by conformal images of positive Killing forms.

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