The sharp curl-Sobolev inequality

Abstract

We solve a longstanding problem, going back at least to Rivi\`ere 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 $\alpha$, the conformally invariant quotient satisfies (with positive denominator) \[ \frac{\Big(\int_{\mathbb{S}^n}|{\rm curl}\alpha|^{\frac{2n}{n+1}}\,{\rm dV}\Big)^{\frac{n+1}{n}}}{\int_{\mathbb{S}^n}\langle{\rm curl}\alpha,\alpha\rangle\,{\rm dV}} \ge \frac{n+1}{2}\,\omega_n^{\frac1n}. \] 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 $\mu([g_{{\rm st}}])$. 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 $\pi:\mathbb{S}^3\to\mathbb{S}^2$ are exactly $\pi\circ\Phi$ with $\Phi\in{\rm Conf}^+(\mathbb{S}^3)$, confirming a conjecture of Rivi\`ere. Third, for the Faddeev-Skyrme energy $\mathcal{FS}_\rho$ 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 $\|{\rm curl} A\|_{3/2} \ge 3\omega_3^{\frac 2 3}$ 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 $n=3$.