On the sharp constants in curl-Sobolev inequalities on \mathbbSⁿ

2026/07/21 by Guofang Wang, Mingwei Zhang
#math.DG #math.AP #math.SP

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Abstract

Let n≡ 3 (mod 4) and set p=(n-1)/(2). On an oriented Riemannian n-manifold we consider the (middle-degree) curl operator, curl:*d:Ω^p→Ω^p, and the associated conformally invariant Sobolev quotients on (\mathbbSⁿ,g_st), J₁(α)=\frac(∫|curlα|^(2n)/(n+1) dV)^(n+1)/(n)∫\langlecurlα,α⟩ dV, J₂(α)=\frac(∫|curlα|^(2n)/(n+1) dV)^(n+1)/(n)inf_ϕ(∫|α-dϕ|^(2n)/(n-1) dV)^(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 J₁ around this family, which in particular implies that every such form is a strict local minimizer in the conformally invariant space W^1,(2n)/(n+1). In contrast, we show that these critical points are unstable for J₂ (and for related conformally invariant quotients), yielding a strict upper bound for the sharp constant of the J₂ inequality. By conformal invariance, the results on \mathbbSⁿ transfer naturally to ℝⁿ.

Citations

• Conformal Killing forms on Riemannian manifolds

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