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Systolic inequalities on the sphere from symplectic embeddings

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Pith's one-line read Fiberwise balanced metrics on the sphere satisfy a systolic inequality via symplectic embeddings.

desk verdict Core balanced systolic inequality is new and correct, but Theorem 3.8's pinching-to-balancing claim is false because the balancing quantities are not isometry-invariant. read the letter →

arxiv 2506.07674 v1 pith:MFIKWLIZ submitted 2025-06-09 math.SG math.DG

Brayan Ferreira This is my paper
classification math.SGmath.DG MSC 53D4253C2353C22
keywords systolicinequalitysymplecticcapacitiesReeborbitsfiberwisestar-shapedhypersurfacebeta-balancedmetricroundspherepinchedcurvaturecotangentbundle
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 that the length of the shortest closed geodesic on a Riemannian two-sphere is controlled by the metric's area whenever the unit co-sphere bundle is fiberwise balanced between two cotangent radii. Concretely, if a metric $g$ satisfies $(r/R)^2 \geq \beta$, where $r$ and $R$ are the smallest and largest cotangent lengths on the unit co-sphere bundle, then $L_{\min}(g)^2 \leq (\pi/\beta)\mathrm{Area}(S^2,g)$. For $\beta \geq \pi/32$ this beats the best universal bound known, and it also yields bounds in terms of the first Laplacian eigenvalue and, under nonnegative curvature, the diameter. The paper also shows that every antipodally symmetric $\delta$-pinched metric is fiberwise $\beta$-balanced for an explicit exponentially small $\beta$, so the main inequality applies to a broad class of positively curved metrics. The proofs work by comparing fiberwise star-shaped hypersurfaces in $T^*S^2$ with round cotangent balls using monotonicity and spectrality of the recently defined symplectic capacities.

What carries the argument

The mechanism is the first symplectic capacity $c_1(X,\omega)$ for four-dimensional Liouville domains, defined by a minimax of pseudoholomorphic curve energies and satisfying monotonicity, conformality, spectrality, and explicit values on balls and round cotangent bundles. The argument uses the chain $A_{\min}(Y,\lambda) \leq c_1(X,\omega) \leq c_1(D^*_{g_0}(R)S^2)=2\pi R$, valid whenever $(X,\omega)$ embeds symplectically into the round cotangent ball; the volume obstruction then converts the radius ratio $R/r$ into an area ratio, producing the factor $1/\beta$. For the pinching statement, the load-bearing analytic tools are an antipodal Onofri-type inequality and explicit Green's function estimates for the round sphere, which control the conformal factor of a $\delta$-pinched metric in terms of $\delta$.

What would settle it

Compute $c_1(X_\Sigma,\omega_{\mathrm{can}})$ for an explicit fiberwise star-shaped $\Sigma \subset T^*S^2$ by a direct embedded-contact-homology calculation and compare it with $A_{\min}(\Sigma,\lambda|_\Sigma)$; finding any $\Sigma$ with $c_1(X_\Sigma) < A_{\min}(\Sigma,\lambda|_\Sigma)$ would falsify the spectrality input and the chain leading to the systolic bound. Alternatively, an explicit fiberwise $\beta$-balanced metric $g$ with $L_{\min}(g)^2 > (\pi/\beta)\mathrm{Area}(S^2,g)$ would directly refute the main theorem.

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Extended reading notes

Core claim

The central claim is an upper bound on the systole in terms of area plus a balancing parameter: for any Riemannian metric $g$ on $S^2$ whose unit co-sphere bundle $S^*_g(1)S^2$ satisfies $(r(g)/R(g))^2 \geq \beta$, one has $L_{\min}(g)^2 \leq (\pi/\beta)\mathrm{Area}(S^2,g)$. This is deduced from a more general statement about hypersurfaces: if $\Sigma \subset T^*S^2$ is fiberwise star-shaped and $\beta$-balanced, then $A_{\min}(\Sigma,\lambda|_\Sigma)^2 \leq \mathrm{Vol}(\Sigma,\lambda|_\Sigma)/(2\beta)$. For the round metric both inequalities are equalities, and for ellipsoid metrics the balancing parameter coincides with the square root of the curvature pinching ratio. A second theorem states that antipodally symmetric $\delta$-pinched metrics are fiberwise $\beta$-balanced with an explicit exponentially small $\beta$, given as a closed expression in $\delta$.

Load-bearing premise

The proof rests on a theorem imported from [Hut22] stating that the smallest capacity $c_1(X,\omega)$ of a four-dimensional Liouville domain is realized as the action of a periodic Reeb orbit (or orbit set) on its boundary; if that spectrality property failed for the fiberwise star-shaped domains studied here, the inequality $A_{\min} \leq c_1$ would not follow and the systolic bounds would lose their foundation.

Editorial extensions

If this is right

  • For any fiberwise $\beta$-balanced metric on $S^2$, $L_{\min}(g)^2 \leq (\pi/\beta)\mathrm{Area}(S^2,g)$; for $\beta \geq \pi/32$ this improves the universal bound $L_{\min}(g)^2 \leq 32\,\mathrm{Area}(S^2,g)$.
  • Combining with Hersch's eigenvalue bound gives $L_{\min}(g)^2 \leq 8\pi^2/(\beta\,\lambda_1(S^2,g))$, so balancing controls the systole through the first Laplace eigenvalue.
  • Under nonnegative curvature, $L_{\min}(g) \leq 2\sqrt{2}\,\beta^{-1/2}\,D(S^2,g)$, which improves on the known diameter bound when $\beta \geq 8/9$.
  • The explicit $\beta(\delta)$ from Theorem 3.8 turns the main inequality into a systolic bound for every antipodally symmetric $\delta$-pinched metric.
  • The same systolic inequality holds for Finsler metrics when the Holmes–Thompson area is used in place of the Riemannian area.

Reading between the lines

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

  • One could hope to replace the antipodal symmetry in Theorem 3.8 by weaker symmetry or curvature assumptions with explicit constants, extending the same embedding argument to a wider class of metrics.
  • The proof uses only the inclusion $X_\Sigma \subset (R/r)\,X_\Sigma$; any fiberwise star-shaped $\Sigma$ for which a tighter symplectic embedding into a dilated copy exists would sharpen the constant, and toric domain results suggest such cases among metrics of revolution.
  • A numerical study of $c_1(D^*_g(1)S^2)$ versus $L_{\min}(g)$ for explicit metrics such as dumbbells, ellipsoids, and surfaces of revolution could quantify how far the capacity method sits from the sharp systolic inequality, complementing the paper's observation that equality fails in those families.
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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 / 2 minor

Summary. The paper uses Hutchings' symplectic capacities to prove upper bounds on the minimal Reeb action of fiberwise star-shaped hypersurfaces in T^*S^2, introduces a notion of fiberwise β-balanced hypersurfaces, and derives systolic inequalities for Riemannian metrics on S^2. The main results are: Theorem 2.1 bounding Amin by 2πR(Σ), Theorem 2.3 bounding Amin^2 by Vol/(2β) for β-balanced hypersurfaces, Theorem 3.1 giving Lmin(g)^2 ≤ (π/β)Area(S^2,g) for fiberwise β-balanced metrics, and Theorem 3.8 claiming that antipodally symmetric δ-pinched metrics are fiberwise β-balanced for an explicit exponentially small β. The paper also contains a checkable, explicit-constant estimate of the conformal factor via the Onofri inequality.

Significance. If the main results held, the paper would provide a clean symplectic-capacity route to systolic inequalities with explicit constants, and the bridge from a balancing condition to the systole in Theorem 3.1 is elegant and independently useful. The derivations of Theorems 2.1 and 2.3 are short, transparent, and largely checkable, and the use of the spectrality and monotonicity of Hutchings' capacities is clearly explained. However, the advertised pinching-to-balancing result (Theorem 3.8) is false as stated, because the balancing condition is not invariant under the isometries introduced by Uniformization. This undermines the claimed applications to arbitrary antipodally symmetric δ-pinched metrics, although the earlier systolic inequalities for β-balanced metrics remain valid.

major comments (2)
  1. [§3.3, proof of Theorem 3.8] Theorem 3.8 is false as stated. Fiberwise β-balancing is defined using the fixed round metric g0 in the fixed cotangent bundle T^*S^2, i.e. R(g) and r(g) depend on the given metric g, not on its isometry class. The proof replaces g by a conformal representative e^{2u}g0 via Uniformization and writes 'We write K_{e^{2u}g0}=K_g, omitting the uniformization isometry.' This omission is not legitimate: an isometry preserves curvature and the systole, but it does not preserve R(g), r(g), or β. Concretely, let ψ be an antipodal-equivariant diffeomorphism of the form ψ(θ,ϕ)=(F(θ),ϕ) with F(π−θ)=π−F(θ), and put g=ψ^*g0. Then g has constant curvature 1, so it is δ-pinched with δ=1, and it is antipodally symmetric. Direct computation gives R(g)^2=max(F'(θ)^2, sin^2F(θ)/sin^2θ) and r(g)^2=min(F'(θ)^2, sin^2F(θ)/sin^2θ). Choosing F with derivative M on a tiny interval and derivative close to ε elsewhere makes R/r arbitrarily large, hence β=(r/R)^2 arbitrarily small, contradicting the fixed positive lower bound claimed for δ=1. Thus the theorem, and the subsequent claim that every δ-pinched antipodal metric satisfies the stated β-bound, cannot be correct.
  2. [§3.3, proof of Theorem 3.8] The sentence 'Since g is antipodally symmetric, we can assume u(q)=u(−q)' is not justified. For a conformal metric e^{2u}g0, antipodal symmetry of the metric implies u(−q)=u(q), but when g is merely isometric to e^{2u}g0 through an arbitrary uniformizing diffeomorphism, antipodal symmetry of g does not force the conformal factor in that representation to be antipodally symmetric. An equivariant uniformization statement would be needed, or the theorem must be restricted to conformal representatives. This is a second, independent gap in the same proof.
minor comments (2)
  1. [Appendix, Lemma A.2] The first nonzero Laplacian eigenvalue on the round sphere is λ1(g0)=2, not 1/2. The displayed Poincaré inequality ∥u0∥^2_{L^2} ≤ (1/2)∥∇u0∥^2_{L^2} is correct because the reciprocal of the eigenvalue is 1/2, but the parenthetical identification of the eigenvalue is wrong and should be corrected.
  2. [§3.1.1] The identity Vol(D^*_g(1)S^2, ω_can)=2π Area(S^2,g) is used without proof or citation. A brief derivation or a reference (e.g., the standard Liouville measure computation for unit cotangent bundles) would make the argument self-contained.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the systolic bounds follow from external Hutchings capacity properties and elementary volume/monotonicity arguments; the self-citations are not load-bearing.

full rationale

The core derivation is conditional and non-circular. Theorem 1.2 is a direct consequence of monotonicity, conformality, and spectrality of Hutchings' c1, all imported from [Hut22]; no quantity in the paper is fitted to make an inequality true. Theorem 2.3 follows by combining the inclusion XΣ ⊂ D*_g0(R(Σ))S2, the volume obstruction, and the definition of β-balanced; β is a free variable in a definition, not a parameter fitted to the target systolic bound. Theorem 3.1 then uses only the identity Vol(D*_g(1)S2)=2πArea(S2,g). The paper's self-citations, [FR22] and [Fer24], occur in background discussions: the round-metric capacity c1(D*_g0(1)S2)=2π is quoted from [Hut22, Theorem 17], and the [Fer24] dumbbell remark is not used in any proof. The proof of Theorem 3.8 contains a possible gap where an isometric conformal representative is identified with the original metric ('We write K_{e^{2u}g0}=K_g, omitting the uniformization isometry'), which is a correctness concern rather than a circularity: the derived β(δ) is produced by independent analytic inequalities (Lemma 3.5, Lemma 3.4, Appendix A) and is not a restatement of the input. No equation in the paper equals another by construction, and no fitted input is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 10 assumptions · 0 invented entities

No numerical parameters are fitted to data. The arguments rest on external theorems from Hutchings, OPS88, Aubin, and standard symplectic and geometric facts; these are listed as axioms. The paper introduces the new definition 'fiberwise β-balanced' but no new objects of the graviton type.

assumptions (10)
  • standard math Hutchings capacity properties (Theorem 1.1): monotonicity, conformality, spectrality, and the ball/round metric values for c_k.
    Imported from [Hut22, Thm 6]; supplies the inequality Amin <= c1(X) used in Theorems 1.2, 2.1 and 2.3.
  • standard math There are symplectic embeddings intP(2π,2π) → D*_g0(1)S2 → S2×S2.
    From [FR22, Thm 1.1] and [OU16, Lemma 2.3] as stated in the introduction; used to fix c1(D*_g0(1)S2)=2π.
  • standard math Volume of the unit disk cotangent bundle of (S2,g) satisfies Vol(D*_g(1)S2,ωcan)=2πArea(S2,g).
    Used without proof in Theorem 3.1 to turn Theorem 2.3 into a metric systolic bound; standard Liouville volume computation.
  • standard math Volume obstruction for symplectic embeddings: a symplectic embedding (A,ω)→(B,σ) implies ∫Aω^2 ≤ ∫Bσ^2.
    Used in the proof of Theorem 2.3; basic symplectic fact.
  • standard math OPS88 Corollary 2.2: for mean-zero antipodally symmetric u, ln(∫e^u dσ0) ≤ (1/8)∫|∇u|^2 dσ0.
    Central to Lemma 3.5's gradient bound and hence to Theorem 3.8.
  • standard math Uniformization theorem: every metric on S2 is isometric to e^{2u}g0.
    Used in Section 3.1 and Theorem 3.8 to place metrics in conformal form.
  • standard math Gauss-Bonnet theorem: ∫ K_g dA_g = 4π.
    Used in Lemma 3.7 and Lemma 3.5 to normalize integrals of K_g e^{2u}.
  • standard math Green's function representation and lower bound for the round sphere Laplacian.
    Used in Lemma 3.7 to prove min u ≥ ar u - 1.
  • standard math Sobolev embedding H2(S2) → C0(S2) with the explicit spherical harmonic constant from Lemma A.1.
    Used in Theorem 3.8 to bound oscillation by ∥Δu∥_{L2}.
  • domain assumption For a fiberwise star-shaped hypersurface Σ ⊂ T*S2, the enclosed region XΣ is a Liouville domain with contact boundary (Σ, λ|Σ).
    Definitional assumption stated in Section 2; underlies the application of capacity spectrality.

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Pith. "Pith review of Systolic inequalities on the sphere from symplectic embeddings." pith.science (2026). https://pith.science/paper/MFIKWLIZ

@misc{pith2026250607674,
  author       = {Pith},
  title        = {Pith review of: Systolic inequalities on the sphere from symplectic embeddings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MFIKWLIZ}},
  note         = {Machine review of arXiv:2506.07674}
}
abstract

We use properties of symplectic capacities that were recently defined by Hutchings to obtain upper bounds on the minimal action of Reeb orbits on fiberwise star-shaped hypersurfaces $\Sigma \subset T^*S^2$. In addition, we introduce the notion of a fiberwise $\beta$-balanced hypersurface $\Sigma \subset T^*S^2$ and establish upper bounds for the systole in terms of $\beta$ and geometric data, in the case of Riemannian metrics on $S^2$ satisfying this property. Finally, under the assumption of antipodal symmetry, we provide a non-sharp estimate of how fiberwise balanced a $\delta$-pinched metric is.

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