AI 中文总结
本文研究了$\mathbb{S}^n$上curl-Sobolev不等式中精确常数的问题，证明了$J_1$的局部稳定性及$J_2$的不稳定性，给出了精确常数的严格上界。
AI 中文摘要
设$n\equiv 3\ (\mathrm{mod}\ 4)$，并设$p=\frac{n-1}{2}$。在定向的Riemannian $n$-流形上考虑（中间次数）curl算子，$\mathrm{curl}:*\mathrm{d}:\Omega^{p}\rightarrow\Omega^{p}$，以及在$(\mathbb{S}^n,g_{\mathrm{st}})$上的相关 conformally 不变的Sobolev商，$$ J_1(\alpha)=\frac{\big(\int|\mathrm{curl}\alpha|^{\frac{2n}{n+1}}\,\mathrm{dV}\big)^{\frac{n+1}{n}}}{\int\langle\mathrm{curl}\alpha,\alpha\rangle\,\mathrm{dV}}, \qquad J_2(\alpha)=\frac{\big(\int|\mathrm{curl}\alpha|^{\frac{2n}{n+1}}\,\mathrm{dV}\big)^{\frac{n+1}{n}}}{\inf_{\phi}\big(\int|\alpha-\mathrm{d}\phi|^{\frac{2n}{n-1}}\,\mathrm{dV}\big)^{\frac{n-1}{n}}}. $$ Killing $p$-forms及其conformal图像构成了两个函数的自然临界点家族，类似于经典的Sobolev不等式中的Aubin-Talenti家族。我们证明了对于$J_1$在该家族附近的定量局部稳定性估计，这特别意味着每个这样的形式在conformally不变空间$W^{1,\frac{2n}{n+1}}$中都是严格的局部极小值。相反，我们显示这些临界点对于$J_2$（以及相关的conformally不变商）是不稳定的，从而得到$J_2$不等式精确常数的严格上界。通过conformally不变性，$\mathbb{S}^n$上的结果自然地转移到$\mathbb{R}^n$。
英文摘要
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, $\mathrm{curl}:*\mathrm{d}:Ω^{p}\rightarrowΩ^{p}$, and the associated conformally invariant Sobolev quotients on $(\mathbb{S}^n,g_{\mathrm{st}})$, \[ J_1(α)=\frac{\big(\int|\mathrm{curl}α|^{\frac{2n}{n+1}}\,\mathrm{dV}\big)^{\frac{n+1}{n}}}{\int\langle\mathrm{curl}α,α\rangle\,\mathrm{dV}}, \qquad J_2(α)=\frac{\big(\int|\mathrm{curl}α|^{\frac{2n}{n+1}}\,\mathrm{dV}\big)^{\frac{n+1}{n}}}{\inf_ϕ\big(\int|α-\mathrm{d}ϕ|^{\frac{2n}{n-1}}\,\mathrm{dV}\big)^{\frac{n-1}{n}}}. \] 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 $\mathbb{R}^n$.
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