Critical curl extremals and the hyperbolic plane

Abstract

We study the critical Sobolev-curl quotient on $\mathbb{R}^3$ at the conformally invariant exponent $p=3/2$. Its sharp constant and optimizers were posed as open problems in the literature. We identify the toroidal and poloidal extremals. Two structural facts are used. First, the critical curl--curl equation is self-dual: the map $u\mapsto|\nabla\times u|^{-1/2}\nabla\times u$ is an involution on finite-energy solutions. It is the self-dual case of a duality between the exponents $p$ and $3p/(4p-3)$, which exchanges toroidal and poloidal fields and relates the sharp constants. Second, the toroidal minimization problem reduces exactly to a subcritical $3/2$-Laplace problem on the hyperbolic plane. Combined with symmetry and shooting results, these facts show that the two restricted sharp constants coincide. They also classify the gauge-minimal extremals: these are unique up to scaling and the Euclidean transformations realizing hyperbolic isometries, semi-explicit, and dual to each other. For nonzero one-signed profiles radial about a fixed hyperbolic centre, a gauge-canonical inverse of the duality yields an iteration by one-dimensional quadratures: the quotients are nonincreasing and converge to the restricted sharp constant, while the normalized iterates converge to the extremal pair. The first step from an explicit competitor gives the new upper bound $96\sqrt{15}π/(5\sqrt{4455π-13568})\approx11.2948$ for the restricted constants. The unrestricted sharp constant remains open. As an application, bounded axisymmetric poloidal potentials admit no $L^2$ zero mode of the Weyl-Dirac operator. An explicit conformal transformation extends this exclusion to both extremals at every real coupling.