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

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 $\mathbb{R}^n$.