The sharp curl-Sobolev inequality

Abstract

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 $n\equiv 3(\mathrm{mod}4)$: for every $\frac{n-1}{2}$-form $\alpha$, 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 $\mu ([g_{\mathrm{st}}])$. Second, we prove that the unique minimizers of the $3$-energy ${\int }_{{\mathbb{S}}^3}|\mathrm{d}u{|}^3$ in the homotopy class of the Hopf map $\pi :{\mathbb{S}}^3\to {\mathbb{S}}^2$ are exactly $\pi \circ \Phi$ with $\Phi \in {\mathrm{Conf}}^{+}({\mathbb{S}}^3)$, confirming a conjecture of Rivière. Third, for the Faddeev-Skyrme energy ${\mathcal{F}\mathcal{S}}_{\rho }$ on ${\mathbb{S}}^3$, we establish global minimality of the Hopf map in the full predicted range: for every coupling constant $\rho \le \sqrt{2}$, the unique global minimizers in its homotopy class are precisely $\pi \circ R$ 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 $\Vert \mathrm{curl}A{\Vert }_{3/2}\ge 3{\omega }_3^{\frac{2}{3}}$ 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 $n=3$.

v3: Appendix C added, including the clarification of regularity