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The sharp curl-Sobolev inequality

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Pith's one-line read The paper proves the sharp curl–Sobolev inequality on spheres of dimension 3 mod 4, with equality exactly for positive Killing forms, and derives five consequences in geometry and physics.

arxiv 2607.23827 v2 pith:D4SCVJWH submitted 2026-07-26 math.DG math-phmath.MP

Guofang Wang , Mingwei Zhang This is my paper · ORCID
classification math.DGmath-phmath.MP MSC 35A2346E3558A10
keywords curl–SobolevinequalityKillingformsconformalinvarianceHopfmapp-harmonicFaddeev–Skyrmemodelzeromodespinor
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 proves a sharp inequality for the curl operator on the round n-sphere when n ≡ 3 (mod 4): for every (n-1)/2-form α with positive helicity, the conformally invariant quotient J(α) is at least (n+1)/2 ω_n^{1/n}, and equality holds exactly when α is a positive Killing form up to conformal transformations and closed forms. The result closes a problem open since 1998, including the physically central case n = 3, and by conformal invariance the same inequality holds on Euclidean space. From it the paper derives: the round metric uniquely optimizes the first positive curl eigenvalue in its conformal class; the Hopf map uniquely minimizes the 3-energy in its homotopy class; the Hopf map globally minimizes the Faddeev–Skyrme energy for every coupling ρ ≤ √2; and the magnetic field supporting a Dirac zero mode on S³ satisfies the sharp bound ‖curl A‖_{3/2} ≥ 3 ω_3^{2/3}, with equality characterized by Killing spinors. It also confirms the predicted 32π² constant in the topological lower bound for the classical Faddeev–Skyrme energy on R³.

What carries the argument

The argument rests on three mechanisms. First, the reduction of the conformally invariant quotient to a normalized nonlinear eigenvalue problem, curl α = (n+1)/2 |α|^{2/(n-1)} α. Second, the pointwise orthogonal decomposition ∇β = P + Q + S of the covariant derivative of the unit form β = α/|α|: the pieces are constructed to isolate first-order terms in ∇ log f from zeroth-order terms in f, yielding the sharp Kato-type inequality |∇α|² ≥ ((n+1)/(n-1))|∇f|² + (4/(n-1)²)|∇f⌟β|² + ((n+1)/2) f^{2(n+1)/(n-1)}. Third, the sharp spherical Gagliardo–Nirenberg inequality, which is obtained from the sharp Euclidean Gagliardo–Nirenberg inequality by stereographic projection and averaging with its Kelvi

What would settle it

Compute the quotient J for a concrete non-Killing form on S³, for instance a suitable linear combination of two curl eigenforms of opposite signs, and check whether it falls below 2 ω_3^{1/3}; a counterexample at any value would refute Theorem 1.1. Alternatively, perform the direct first variation of J with respect to α under the constraint ∫⟨curl α, α⟩ > 0 and verify whether the resulting Euler–Lagrange equation is equivalent to (4.1); if the normalized equation fails for a genuine minimizer, the chain of inequalities in Section 4 loses its starting point. The auxiliary bound µ⁻([g_st]) can b

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Extended reading notes

Core claim

The central claim is a sharp curl–Sobolev inequality on the round n-sphere, n ≡ 3 (mod 4). For any (n-1)/2-form α with ∫⟨curl α, α⟩ > 0, the quotient J(α) = (∫|curl α|^{2n/(n+1)})^{(n+1)/n} / ∫⟨curl α, α⟩ is at least (n+1)/2 ω_n^{1/n}. The proof normalizes a minimizer to satisfy the nonlinear eigenvalue equation curl α = (n+1)/2 |α|^{2/(n-1)} α, writes α = f β with |β|=1, and decomposes ∇β into three pointwise orthogonal pieces P, Q, S. This yields a Kato-type inequality bounding |∇α|² from below by (n+1)/(n-1)|∇f|² + (4/(n-1)²)|∇f⌟β|² + (n+1)/2 f^{2(n+1)/(n-1)}. Combining the Bochner–Weitzenböck formula with this inequality gives the integral estimate (n-1)²/4 ∫ f^{2(n+1)/(n-1)} ≥ ∫ |∇f|² +

Load-bearing premise

The load-bearing premise is that a minimizer of the quotient J can be rescaled to satisfy the pointwise Euler–Lagrange equation curl α = (n+1)/2 |α|^{2/(n-1)} α; the proof uses this identity in the Bochner–Weitzenböck step and in the cancellation that yields the key integral estimate, but does not show a derivation from the first variation of J. A secondary fragile premise is the assertion that the negative-side conformal eigenvalue bound µ⁻([g_st]) equals (n+1)/2 ω_n^{1/n},

Editorial extensions

If this is right

  • The round metric on S^n is the unique optimizer of the conformal invariant µ([g]), the infimum of the first positive curl eigenvalue times Vol^{1/n}, over all metrics conformally equivalent to the round metric.
  • The Hopf map from S³ to S² is the unique minimizer of the 3-energy ∫|du|³ in its homotopy class, up to orientation-preserving conformal transformations, confirming a conjecture from 1998.
  • In the Faddeev–Skyrme model on S³, the Hopf map is the unique global minimizer of the energy with coupling ρ ≤ √2, up to rotations of S³; the threshold ρ = √2 is exactly the stability threshold.
  • On S³, any magnetic field supporting a nontrivial Dirac zero mode satisfies the sharp bound ‖curl A‖_{3/2} ≥ 3 ω_3^{2/3}; equality characterizes the pairing of a Killing spinor with its Reeb field, answering which magnetic fields support zero modes.
  • The classical Faddeev–Skyrme energy on R³ obeys the sharp topological lower bound E(u) > 32π² |Q(u)|^{3/4}, confirming the conjectured value of the constant; for Hopf number 1 the actual infimum is strictly larger than 32π².

Reading between the lines

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

  • The same normalization strategy should extend to n ≡ 1 (mod 4) by using the self-adjoint curl defined with an imaginary unit on complex forms, giving the analogous sharp inequality on the sphere and on R^n.
  • The companion conformally invariant functional I(α) — where the denominator is replaced by the distance to closed forms — has no minimizer at the conjectured constant; the failure suggests the optimal constant for that functional differs from (n+1)²/4 ω_n^{2/n}, and its determination remains open.
  • The Yamabe-type problem for forms posed in the paper — whether strict inequality in Y_{(n-1)/2}(M,[g]) ≤ Y_{(n-1)/2}(S^n) holds for every non-conformally-spherical manifold — is the natural next test of the method; a proof would yield existence of minimizers on all such manifolds.
  • The spinor–vector identification used in the zero-mode proof (which is special to dimension 3) hints that a higher-dimensional analog of the sharp bound for ‖curl A‖ would require a different decomposition; the paper states the expectation that the strategy extends for n ≥ 3.
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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

2 major / 3 minor

Summary. The paper claims a sharp conformally invariant curl–Sobolev inequality on S^n for n≡3 mod 4, with the sharp constant (n+1)/2 ω_n^{1/n} and equality precisely for positive Killing (n-1)/2-forms modulo orientation-preserving conformal transformations and ker(d). The proof is structured around a new sharp spherical Gagliardo–Nirenberg inequality, a decomposition of ∇β into orthogonal components, a Kato-type identity, and the Bochner–Weitzenböck formula. The authors then derive four main applications: optimal conformal metrics for the first positive curl eigenvalue on S^n, global minimality of the Hopf map for the 3-energy and for the Faddeev–Skyrme energy for ρ≤√2, and a sharp lower bound for ‖curl A‖_{3/2} for Dirac zero modes, together with a Ward-type bound in Appendix B.

Significance. If correct, Theorem 1.1 would settle a problem open since Rivière 1998 and would immediately imply several well-known conjectures in geometry and mathematical physics. The paper has notable strengths: the spherical Gagliardo–Nirenberg inequality is proved carefully, including the delicate n=2 boundary correction; the constants are explicit; the conformal-invariance framework is coherent; and the downstream algebra from (4.1) through Lemma 4.9 to (4.12) is checkable. However, the central variational reduction to (4.1) is not derived and, as stated, appears to be an equation for the wrong variable. Because this step is the input to all the later cancellation, the main theorem is not established in the present form. The significance of the paper is therefore conditional on a substantial repair of the proof.

major comments (2)
  1. [§4, Eq. (4.1)] The reduction to (4.1) is the load-bearing step and is not justified. For a critical point of J, direct variation gives curl(|curlα|^{-2/(n+1)}curlα) = (F/D) curlα, where F=∫|curlα|^{2n/(n+1)} and D=∫⟨curlα,α⟩. Setting β=|curlα|^{-2/(n+1)}curlα yields curlβ = (F/D)|β|^{2/(n-1)}β. This is an equation for β, not for α. Adding a representative in ker(d) does not change curlα, and rescaling β does not make the original α satisfy (4.1). The argument can be repaired by introducing the rescaled dual form γ=sβ satisfying (4.1) and proving J(α)=(n+1)/2(∫|γ|^{2n/(n-1)})^{1/n}, but this dual-variable change is not stated, and all subsequent identities (4.3)–(4.11) are written for α. The transfer of the equality case back to α is also not explained. Without this derivation, Theorem 1.1 is unproved.
  2. [§5, Remark 5.2 and §8, Eq. (8.26)] The proof of Theorem 1.6 uses both eigenvalue bounds in (8.26). The negative-side bound µ⁻([g_st])=(n+1)/2 ω_n^{1/n} is asserted in Remark 5.2 by 'The same argument implies' with no proof. Theorem 1.1 concerns only positive helicity, and for a general curl operator the spectrum need not be symmetric, so the negative eigenvalue bound is not immediate. Since (8.26) is used directly to prove claim (8.25), which is essential for Theorem 1.6, this assertion must either be proved in detail or the argument must be modified to avoid it.
minor comments (3)
  1. [§5, Lemma 5.1] The text says Theorem 1.3 is 'in fact equivalent' to Theorem 1.1, but only the direction needed for the applications is proved. If equivalence is claimed, the reverse implication should be stated and proved, or the wording should be softened.
  2. [§8.3] The regularization of the zero-mode proof is sketched and the equality classification is deferred to [17]. Since the equality case in Theorem 1.6 is one of the main results, the list of equality conditions (1)–(6) should be verified after the approximation argument, not only in the smooth case.
  3. [§3, Step 1] In the n=2 case, the correction term in (3.11) is expressed in terms of f(N), the value at the north pole, which is not a conformally invariant quantity. The final equality is restored after combining with (3.12), but a short remark explaining the cancellation would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the sharp curl–Sobolev inequality is derived from an external GN inequality plus in-paper Bochner/Kato estimates, with the sharp constant emerging from the matching constants rather than being used as an input.

full rationale

The derivation chain in Section 4 is genuinely first-principles relative to the stated inputs. A minimizer (existence cited from Rivière [36]) is normalized via the claimed Euler–Lagrange/scaling reduction (4.1); the normalization has an explicit missing derivation—the direct first variation of J gives curl(|curl α|^{-2/(n+1)} curl α) = const·curl α, not (4.1)—so the assertion is a correctness gap, but it is not a circular reduction: (4.1) is not the target inequality, and the paper does not fit a parameter to the target constant. From (4.1), the quotient becomes ((n+1)/2)(∫|α|^{2n/(n-1)})^{1/n}, and the proof then reduces to proving ∫|α|^{2n/(n-1)} ≥ ω_n. That reduction is accomplished by the Bochner–Weitzenböck identity, the pointwise orthogonal decomposition of ∇β, the Kato-type lower bound (4.5), and Lemma 4.9, combined with the spherical Gagliardo–Nirenberg inequality (Theorem 3.2). Theorem 3.2 is proved in-paper from the external Del Pino–Dolbeault inequality [8] via stereographic projection and a Kelvin-transform averaging argument; it is independent of the curl–Sobolev statement. The sharp constant (n+1)/2 ω_n^{1/n} is obtained when the coefficient (n+1)/2 from (4.1) multiplies the scalar GN constant ω_n^{2/n}; the equality analysis then verifies that Killing forms attain it. Thus the equality case is a check of sharpness, not an input. The applications are consequences or equivalences (Lemma 5.1, Sections 6–8), not circular dependencies. The only self-citations ([42], [43], [45]) are peripheral and not load-bearing for Theorem 1.1. Remark 5.2's µ^- value is asserted by 'the same argument' without proof and is used in Theorem 1.6, but it is an omitted-support issue, not a circular reuse of the target theorem. Therefore no specific circular step can be exhibited: the theorem is not equivalent to a fitted quantity or to a self-citation chain.

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

The paper uses no fitted parameters: the constant (n+1)/2 ω_n^{1/n} is pinned down by evaluating J on unit-norm positive Killing forms, then proven optimal by a genuine chain of inequalities. The main unproved inputs are external theorems (Del Pino–Dolbeault, Rivière's existence/regularity, standard Bochner/Schrödinger–Lichnerowicz identities), the asserted variational reduction to (4.1), and the asserted spectral statement in Remark 5.2. No new entities are postulated.

assumptions (6)
  • standard math Sharp Euclidean Gagliardo–Nirenberg inequality of Del Pino–Dolbeault (Theorem 3.1) with constant (n-1)²/4 ω_n^{2/n} and extremal family (3.2).
    External theorem; base of Theorem 3.2 via Kelvin transform + averaging + stereographic pullback. Load-bearing for the final constant match.
  • domain assumption Existence and C^{1,γ} regularity of a minimizer of J among forms with ∫⟨curlα,α⟩>0, cited from Rivière [36, Prop. IV.1] and Isobe [23].
    The whole proof runs through a minimizer α; the paper does not re-prove existence or regularity, and Section 1 declares regularity out of scope.
  • domain assumption Euler–Lagrange reduction: after gauge choice and rescaling, a minimizer satisfies curl α = (n+1)/2 |α|^{2/(n-1)} α (eq. 4.1).
    Asserted in one sentence ('By combining the Euler–Lagrange equation with the scaling invariance of J...'), not derived. All subsequent integral identities (4.9) and the cancellation in Lemma 4.9 depend on the exact coefficient.
  • standard math Sharp critical Sobolev inequality on S^n (eq. 2.5); Bochner–Weitzenböck identity (4.8); Schrödinger–Lichnerowicz formula (8.5).
    Standard tools. (2.5) is reused in Theorem 1.6 via (8.22); (4.8) is the core of Lemma 4.9; (8.5) underlies the zero-mode estimate.
  • ad hoc to paper µ⁻([g_st]) = (n+1)/2 ω_n^{1/n} (Remark 5.2), asserted by 'The same argument implies', used through (8.26) to prove claim (8.25).
    Unproven negative-eigenvalue analog of Theorem 1.3; the remark itself notes the curl spectrum need not be symmetric, so |λ⁻₁| = λ⁺₁ is not automatic. Load-bearing for Theorem 1.6.
  • domain assumption Pointwise identities extend across the zero set {f=0} of |φ| via Lipschitz regularity (Remark 4.1 and §8.3); division by f is handled by f_ε = √(f²+ε²) regularization.
    For Theorem 1.1 the extension is standard (∇α, ∇f = 0 a.e. on the zero set); for Theorem 1.6 the definition α := fξ^♭ on {f=0}, dα = 0 a.e. there, and the equality-case classification are asserted, with details deferred to [17].

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Pith. "Pith review of The sharp curl-Sobolev inequality." pith.science (2026). https://pith.science/paper/D4SCVJWH

@misc{pith2026260723827,
  author       = {Pith},
  title        = {Pith review of: The sharp curl-Sobolev inequality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D4SCVJWH}},
  note         = {Machine review of arXiv:2607.23827}
}
abstract

We solve a longstanding problem, going back at least to Rivi\`ere 1998 and open even in the physically most relevant case $n=3$, by proving a sharp curl-Sobolev inequality on $\mathbb{S}^n$ when $n\equiv 3\pmod 4$: for every $\frac{n-1}{2}$-form $\alpha$, the conformally invariant quotient satisfies (with positive denominator) \[ \frac{\Big(\int_{\mathbb{S}^n}|{\rm curl}\alpha|^{\frac{2n}{n+1}}\,{\rm dV}\Big)^{\frac{n+1}{n}}}{\int_{\mathbb{S}^n}\langle{\rm curl}\alpha,\alpha\rangle\,{\rm dV}} \ge \frac{n+1}{2}\,\omega_n^{\frac1n}. \] We also classify all extremals in terms of Killing forms. By conformal invariance, the same result holds on $\mathbb{R}^n$. We then give geometric and variational applications that settle several open conjectures in geometry and mathematical physics. First, we show that on $\mathbb{S}^n$ the round metric is the unique optimizer for the conformal invariant $\mu([g_{{\rm st}}])$. Second, we prove that the unique minimizers of the $3$-energy $\int_{\mathbb{S}^3}|{\rm d} u|^3$ in the homotopy class of the Hopf map $\pi:\mathbb{S}^3\to\mathbb{S}^2$ are exactly $\pi\circ\Phi$ with $\Phi\in{\rm Conf}^+(\mathbb{S}^3)$, confirming a conjecture of Rivi\`ere. Third, for the Faddeev-Skyrme energy $\mathcal{FS}_\rho$ on $\mathbb{S}^3$, we establish global minimality of the Hopf map in the full predicted range: for every coupling constant $\rho\le \sqrt{2}$, the unique global minimizers in its homotopy class are precisely $\pi\circ R$ with $R\in\mathrm{SO}(4)$, as expected since Ward 1999. Fourth, in the presence of Dirac zero modes on $\mathbb{S}^3$, we prove the sharp lower bound $\|{\rm curl} A\|_{3/2} \ge 3\omega_3^{\frac 2 3}$ for the magnetic field and characterize equality in terms of Killing spinors; in particular, this yields a sharp criterion for the existence of zero modes and answers a question of Frank-Loss for $n=3$.

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Works this paper leans on

45 extracted references · 1 canonical work pages

  1. [1]

    B. Ammann. A variational problem in conformal spin geometry.Habilitationsschrift, Universit¨ at Hamburg(2003)

    2003

  2. [2]

    Aratyn, L

    H. Aratyn, L. A. Ferreira, and A. H. Zimerman. Exact static soliton solutions of (3 + 1)- dimensional integrable theory with nonzero Hopf numbers.Physical review letters83.9 (1999), p. 1723

    1999

  3. [3]

    C. B¨ ar. The curl operator on odd-dimensional manifolds.J. Math. Phys.60.3 (2019)

    2019

  4. [4]

    Bracchi, M

    G. Bracchi, M. Capoferri, and D. Vassiliev. Higher order Weyl coefficients for the operator curl (2026). arXiv:2607.03273

    arXiv 2026

  5. [5]

    Cantarella, D

    J. Cantarella, D. DeTurck, H. Gluck, and M. Teytel. Isoperimetric problems for the helicity of vector fields and the Biot–Savart and curl operators.J. Math. Phys.41.8 (2000), pp. 5615–5641

    2000

  6. [6]

    J. S. Case. A Yamabe-type problem on smooth metric measure spaces.J. Differ. Geom.101.3 (2015), pp. 467–505

    2015

  7. [7]

    Colbois and A

    B. Colbois and A. El Soufi. Eigenvalues of the Laplacian acting onp-forms and metric conformal deformations.Proc. Amer. Math. Soc.134 (2006), pp. 715–721

    2006

  8. [8]

    Del Pino and J

    M. Del Pino and J. Dolbeault. Best constants for Gagliardo–Nirenberg inequalities and appli- cations to nonlinear diffusions.J. Math. Pures Appl. (9)81.9 (2002), pp. 847–875. REFERENCES 39

    2002

Show all 45 references
  1. [9]

    El Soufi and S

    A. El Soufi and S. Ilias. Immersions minimales, premi` ere valeur propre du laplacien et volume conforme.Math. Ann.275.2 (1986), pp. 257–267

    1986

  2. [10]

    Enciso, W

    A. Enciso, W. Gerner, and D. Peralta-Salas. Optimal convex domains for the first curl eigenvalue in dimension three.Trans. Amer. Math. Soc.377.07 (2024), pp. 4519–4540

    2024

  3. [11]

    Enciso, W

    A. Enciso, W. Gerner, and D. Peralta-Salas. Optimal metrics for the first curl eigenvalue on 3-manifolds.Calc. Var. PDEs64.146 (2025)

    2025

  4. [12]

    M. J. Esteban. A direct variational approach to Skyrme’s model for meson fields.Comm. Math. Phys.105 (1986), pp. 571–591

    1986

  5. [13]

    L. D. Faddeev. Some comments on the many-dimensional solitons.Letters in Mathematical Physics1.4 (1976), pp. 289–293

    1976

  6. [14]

    Figalli and Y

    A. Figalli and Y. R. Y. Zhang. Sharp gradient stability for the Sobolev inequality.Duke Math. J.171.12 (2022), pp. 2407–2459

    2022

  7. [15]

    R. L. Frank and M. Loss. Existence of optimizers in a Sobolev inequality for vector fields.Ars Inven. Anal.1 (2022)

    2022

  8. [16]

    R. L. Frank and M. Loss. Which magnetic fields support a zero mode?J. Reine Angew. Math. 788 (2022), pp. 1–36

    2022

  9. [17]

    R. L. Frank and M. Loss. A sharp criterion for zero modes of the Dirac equation.J. Eur. Math. Soc. (JEMS)28.1 (2026), pp. 305–331

    2026

  10. [18]

    M. Gillard. Hopf solitons in the AFZ model.Nonlinearity24.10 (2011), pp. 2729–2743

    2011

  11. [19]

    Guerra, X

    A. Guerra, X. Lamy, and K. Zemas. Global Minimality of the Hopf Map in the Faddeev–Skyrme Model with Large Coupling Constant.Comm. Math. Phys.407.8 (2026), p. 171

    2026

  12. [20]

    D. Harland. Topological energy bounds for the Skyrme and Faddeev models with massive pions. Phys. Lett. B728 (2014), pp. 518–523

    2014

  13. [21]

    J. Hersch. Quatre propri´ et´ es isop´ erim´ etriques de membranes sph´ eriques homog` enes.C. R. Acad. Sci. Paris S´ er. A–B270 (1970), pp. 1645–1648

    1970

  14. [22]

    T. Isobe. On a minimizing property of the Hopf soliton in the Faddeev–Skyrme model.Rev. Math. Phys.20.07 (2008), pp. 765–786

    2008

  15. [23]

    T. Isobe. Nonlinear Dirac equations with critical nonlinearities on compact spin manifolds.J. Funct. Anal.260.1 (2011), pp. 253–307.doi:10.1016/j.jfa.2010.09.008

    2011 doi

  16. [24]

    P. Jammes. Minoration conforme du spectre du laplacien de Hodge-de Rham.Manuscripta Math.123.1 (2007), pp. 15–23

    2007

  17. [25]

    Julio-Batalla

    J. Julio-Batalla. Spinor inequality for magnetic fields on spin manifolds (2026). arXiv:2603. 23218

    2026

  18. [26]

    Kundu and Y

    A. Kundu and Y. P. Rybakov. Closed-vortex-type solitons with Hopf index.Journal of Physics A: Mathematical and General15.1 (1982), pp. 269–275

    1982

  19. [27]

    H. B. Lawson and M. L. Michelsohn.Spin Geometry. Princeton Mathematical Series. Princeton University Press, 1989.isbn: 9780691085425

    1989

  20. [28]

    Lin and Y

    F. Lin and Y. Yang. Existence of energy minimizers as stable knotted solitons in the Faddeev model.Comm. Math. Phys.249 (2004), pp. 273–303

    2004

  21. [29]

    Lin and Y

    F. Lin and Y. Yang. Energy splitting, substantial inequality, and minimization for the Faddeev and Skyrme models.Comm. Math. Phys.269.1 (2007), pp. 137–152

    2007

  22. [30]

    Loss and H

    M. Loss and H. T. Yau. Stability of Coulomb systems with magnetic fields: III. Zero energy bound states of the Pauli operator.Comm. Math. Phys.104.2 (1986), pp. 283–290

    1986

  23. [31]

    F. C. Marques and A. Neves. Min-max theory and the Willmore conjecture.Ann. of Math179.2 (2014), pp. 683–782

    2014

  24. [32]

    Mederski and A

    J. Mederski and A. Szulkin. Optimizers in Sobolev-curl inequalities (2025). arXiv:2511.01432

    2025

  25. [33]

    S. Montiel. The Isoperimetric Problem for the Curl Operator (2023). arXiv:2307.09556. 40 REFERENCES

    2023 arXiv

  26. [34]

    D. A. Nicole. Solitons with non-vanishing Hopf index.Journal of Physics G: Nuclear Physics 4.9 (1978), pp. 1363–1369

    1978

  27. [35]

    J. Reuß. A note on the existence of nontrivial zero modes on Riemannian manifolds (2025). arXiv:2503.01602

    2025

  28. [36]

    Rivi` ere

    T. Rivi` ere. Minimizing fibrations andp-harmonic maps in homotopy classes fromS 3 intoS 2. Comm. Anal. Geom.6.3 (1998), pp. 427–483

    1998

  29. [37]

    Rivi` ere

    T. Rivi` ere. Harmonic maps fromS 3 intoS 2 with low Morse index.J. Differ. Geom.125.1 (2023), pp. 173–185

    2023

  30. [38]

    J. M. Speight and M. Svensson. On the strong coupling limit of the Faddeev–Hopf model. Comm. Math. Phys.272.3 (2007), pp. 751–773

    2007

  31. [39]

    J. M. Speight and M. Svensson. Some global minimizers of a symplectic Dirichlet energy.Q. J. Math.62.3 (2011), pp. 737–745

    2011

  32. [40]

    Sutcliffe

    P. Sutcliffe. Knots in the Skyrme–Faddeev model.Proceedings of the Royal Society A: Mathe- matical, Physical and Engineering Sciences463.2087 (2007), pp. 3001–3020

  33. [41]

    A. F. Vakulenko and L. V. Kapitansky. Stability of solitons inS 2 in the nonlinearσ-model.Sov. Phys.-Dokl.(Engl. Transl.);(United States)24.6 (1979)

    1979

  34. [42]

    Wang and M

    G. Wang and M. Zhang. Conformal invariants for the zero mode equation (2025). arXiv:2512. 17854

    2025

  35. [43]

    Wang and M

    G. Wang and M. Zhang. On the sharp constants in curl-Sobolev inequalities onS n (2026). arXiv: 2607.19091

    2026 arXiv

  36. [44]

    R. S. Ward. Hopf solitons onS 3 andR 3.Nonlinearity12.2 (1999), pp. 241–246

    1999

  37. [45]

    M. Zhang. A conformal lower bound of weighted Dirac eigenvalues on manifolds with boundary (2026). arXiv:2603.10875. Albert-Ludwigs-Universit¨at Freiburg, Mathematisches Institut, Ernst-Zermelo-Str. 1, D-79104 Freiburg, Germany Email address:guofang.wang@math.uni-freiburg.de W...

    2026

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