AI 中文总结
解决\(n\equiv 3\pmod 4\)时\(\mathbb{S}^n\)上的尖锐旋度-索伯列夫不等式问题，通过证明不等式并分类极值，给出几何和变分应用，解决了几何与数学物理中的多个猜想。
AI 中文摘要
我们解决了一个长期存在的问题，该问题至少可追溯到1998年的里维耶，甚至在物理上最相关的\(n = 3\)的情况下也是开放的。通过证明当\(n\equiv 3\pmod 4\)时在\(\mathbb{S}^n\)上的尖锐旋度-索伯列夫不等式：对于每个\(\frac{n - 1}{2}\)-形式\(\alpha\)，共形不变商（分母为正）满足\[ \frac{\Big(\int_{\mathbb{S}^n}|{\rm curl}\alpha|^{\frac{2n}{n + 1}}\,{\rm dV}\Big)^{\frac{n + 1}{n}}}{\int_{\mathbb{S}^n}\langle{\rm curl}\alpha,\alpha\rangle\,{\rm dV}} \ge \frac{n + 1}{2}\,\omega_n^{\frac{1}{n}}.\]我们还根据克利福德形式对所有极值进行了分类。通过共形不变性，在\(\mathbb{R}^n\)上也有相同结果。然后我们给出了几何和变分应用，解决了几何和数学物理中的几个开放猜想。首先，表明在\(\mathbb{S}^n\)上，标准度量是共形不变量\(\mu([g_{\rm st}])\)的唯一优化器。其次，证明了在霍普夫映射\(\pi:\mathbb{S}^3\to\mathbb{S}^2\)的同伦类中\(3\)-能量\(\int_{\mathbb{S}^3}|{\rm d} u|^3\)的唯一极小值恰好是\(\pi\circ\Phi\)，其中\(\Phi\in{\rm Conf}^+(\mathbb{S}^3)\)，证实了里维耶的一个猜想。第三，对于\(\mathbb{S}^3\)上的法捷耶夫-斯凯尔姆能量\(\mathcal{FS}_\rho\)，我们在整个预测范围内建立了霍普夫映射的全局极小性：对于每个耦合常数\(\rho\le \sqrt{2}\)，在其同伦类中的唯一全局极小值恰好是\(\pi\circ R\)，其中\(R\in\mathrm{SO}(4)\)。第四，在\(\mathbb{S}^3\)上存在狄拉克零模的情况下，我们证明了磁场的尖锐下界\(\|{\rm curl} A\|_{3/2} \ge 3\omega_3^{\frac{2}{3}}\)，并根据克利福德旋量对等式进行了刻画；特别是，这产生了零模存在的尖锐标准，并回答了\(n = 3\)时弗兰克-洛斯的一个问题。
英文摘要
We solve a longstanding problem, going back at least to Rivière 1998 and open even in the physically most relevant case $n=3$, by proving a sharp curl-Sobolev inequality on $\mathbb{S}^n$ when $n\equiv 3\pmod 4$: for every $\frac{n-1}{2}$-form $α$, the conformally invariant quotient satisfies (with positive denominator) \[ \frac{\Big(\int_{\mathbb{S}^n}|{\rm curl}α|^{\frac{2n}{n+1}}\,{\rm dV}\Big)^{\frac{n+1}{n}}}{\int_{\mathbb{S}^n}\langle{\rm curl}α,α\rangle\,{\rm dV}} \ge \frac{n+1}{2}\,ω_n^{\frac1n}. \] We also classify all extremals in terms of Killing forms. By conformal invariance, the same result holds on $\mathbb{R}^n$. We then give geometric and variational applications that settle several open conjectures in geometry and mathematical physics. First, we show that on $\mathbb{S}^n$ the round metric is the unique optimizer for the conformal invariant $μ([g_{\rm st}])$. Second, we prove that the unique minimizers of the $3$-energy $\int_{\mathbb{S}^3}|{\rm d} u|^3$ in the homotopy class of the Hopf map $π:\mathbb{S}^3\to\mathbb{S}^2$ are exactly $π\circΦ$ with $Φ\in{\rm Conf}^+(\mathbb{S}^3)$, confirming a conjecture of Rivière. Third, for the Faddeev-Skyrme energy $\mathcal{FS}_ρ$ on $\mathbb{S}^3$, we establish global minimality of the Hopf map in the full predicted range: for every coupling constant $ρ\le \sqrt{2}$, the unique global minimizers in its homotopy class are precisely $π\circ R$ with $R\in\mathrm{SO}(4)$, as expected since Ward 1999. Fourth, in the presence of Dirac zero modes on $\mathbb{S}^3$, we prove the sharp lower bound $\|{\rm curl} A\|_{3/2} \ge 3ω_3^{\frac 2 3}$ for the magnetic field and characterize equality in terms of Killing spinors; in particular, this yields a sharp criterion for the existence of zero modes and answers a question of Frank-Loss for $n=3$.
Commentsv3: Appendix C added, including the clarification of regularity
