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

Let $n\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$ and set $p=\frac{n-1}{2}$. On an oriented Riemannian $n$-manifold we consider the (middle-degree) curl operator, $\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}:\ast \mathrm{d}:{\mathrm{\Omega }}^p\to {\mathrm{\Omega }}^p$, and the associated conformally invariant Sobolev quotients on $({\mathbb{S}}^n,g_{\mathrm{s}\mathrm{t}})$, $$J_1(\alpha )=\frac{(\int \mathrm{\mid }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\alpha {\mathrm{\mid }}^{\frac{2n}{n+1}}\mathrm{d}\mathrm{V}{)}^{\frac{n+1}{n}}}{\int ⟨\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\alpha ,\alpha ⟩\mathrm{d}\mathrm{V}},J_2(\alpha )=\frac{(\int \mathrm{\mid }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\alpha {\mathrm{\mid }}^{\frac{2n}{n+1}}\mathrm{d}\mathrm{V}{)}^{\frac{n+1}{n}}}{\underset{\phi }{\inf }(\int \mathrm{\mid }\alpha -\mathrm{d}\phi {\mathrm{\mid }}^{\frac{2n}{n-1}}\mathrm{d}\mathrm{V}{)}^{\frac{n-1}{n}}}\mathrm{.}$$ 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 ${\mathbb{R}}^n$.