On comass and stable systolic inequalities

We study the maximum ratio of the Euclidean norm to the comass norm of p-covectors in Euclidean n-space and improve the known upper bound found in the standard references by Whitney and Federer. We go on to prove stable systolic inequalities when the fundamental cohomology class of the manifold is a cup product of forms of lower degree.

💡 Research Summary

The paper investigates the relationship between two natural norms on the space of p‑covectors in Euclidean n‑space: the Euclidean (Hilbert‑Schmidt) norm |·|∗ and the comass norm ∥·∥∗, which is defined as the supremum of the covector’s values on unit simple p‑vectors. The central quantity is the maximal ratio
Cₙ,ₚ = sup_{ϕ≠0} |ϕ|∗ / ∥ϕ∥∗,
which measures how far the comass norm can be from the Euclidean norm. Classical references (Federer, Whitney) give the crude bound Cₙ,ₚ ≤ ⌊n p⌋¹ᐟ². The authors improve this bound substantially.

First, they establish two new recursive inequalities. Proposition 2.3 shows that for 1 < p < n‑1,
C₂ₙ,ₚ ≤ C₂ₙ₋₁,ₚ₋₁ + C₂ₙ₋₁,ₚ.
Arranging the constants C₂ₙ,ₚ in a Pascal‑type triangle and using this recursion yields Corollary 2.4:
C₂ₙ,ₚ ≤ C_{n‑2}^{p‑1}.
Since C_{n‑2}^{p‑1} < Cₙ,ₚ for all non‑trivial p, this already improves the classical estimate.

A more powerful inequality is given in Proposition 2.5: for 1 ≤ k < p < n,
C₂ₙ,ₚ ≤ C_{n‑p‑k}^{k}·C_{p}^{k}·C₂ₙ,ₖ.
Choosing k appropriately (e.g., k = 1) produces bounds that are often dramatically smaller than ⌊n p⌋¹ᐟ². The authors note that the bound is sharp in many low‑dimensional cases.

Next, using calibration theory, they compute exact values of Cₙ,ₚ for several (n,p). They prove:

• C₆,₃ = 2 (via a special 3‑form in ℝ⁶),
• C₈,₄ = √14 (the Cayley 4‑form),
• C₇,₃ = C₇,₄ = √7 (by combining the previous exact values with the recursion).

These exact constants confirm that the inequalities of Proposition 2.5 are essentially optimal.

The paper then turns to wedge products. Proposition 4.1 establishes that for any p‑form ϕ and (n‑p)‑form ψ, |ϕ∧ψ|∗ ≤ C₂ₙ,ₚ ∥ϕ∥∗ ∥ψ∥∗. Corollary 4.2 extends this to arbitrary degrees p,q: ∥ϕ∧ψ∥∗ ≤ C₂_{p+q, p} ∥ϕ∥∗ ∥ψ∥∗, improving earlier bounds that involved the binomial coefficient ⌊p+q p⌋. Proposition 4.3 iterates this estimate to m‑fold wedge products: |ϕ₁∧…∧ϕ_m|∗ ≤ (∏{j=1}^m C₂{j p, p}) ∏_{i=1}^m ∥ϕ_i∥∗. For p = 2 the product reduces to m! as known from previous work.

The final section applies these norm estimates to stable systolic inequalities. The authors introduce the lattice L_p(M) of integral p‑homology classes represented by closed forms with integral periods, and its dual lattice in cohomology. They define a lattice constant Γ_b = sup_{L} λ₁(L)·λ_b(L∗), a variant of the Hermite constant adapted to possibly non‑Euclidean norms.

Theorem 5.2 shows that for any compact orientable n‑manifold M with b‑th Betti number b>0 and any Riemannian metric g, stsys_p(M,g)·stsys_{n‑p}(M,g)·vol(M,g) ≤ C₂ₙ,ₚ·(Γ_b)², where stsys_p denotes the stable p‑systole (the first successive minimum of L_p(M) with respect to the stable mass norm). The proof uses the cup product of a p‑class and an (n‑p)‑class, the pointwise wedge estimate from Proposition 4.1, and the definition of Γ_b.

Theorem 5.4 generalizes this to the situation where the fundamental cohomology class of M can be expressed as a cup product of m classes of degree p. It yields stsys_p(M,g)^m·vol(M,g) ≤ (∏{j=1}^m C₂{j p, p})·(Γ_b)^m. Corollary 5.5 extracts concrete numerical inequalities for dimensions 6, 7, 8 and for 2m‑dimensional manifolds, using the exact constants computed earlier. For instance, in dimension 6 with p = 3, one gets stsys₃(M,g)² ≤ 4·(Γ_b)²·vol(M,g); in dimension 8 with p = 4, stsys₄(M,g)² ≤ 14·(Γ_b)²·vol(M,g); and for CP^m the bound coincides with Gromov’s optimal inequality.

Overall, the paper provides a systematic improvement of the Euclidean‑to‑comass norm ratio, supplies exact constants in several key cases, extends wedge‑product estimates, and translates these analytic improvements into sharper stable systolic inequalities. The work bridges geometric measure theory, calibration geometry, and the geometry of numbers, offering new tools for researchers studying volume‑minimizing cycles, systolic geometry, and related variational problems.