Critical curl extremals and the hyperbolic plane
Abstract
We study the critical Sobolev-curl quotient on ℝ3 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↦|∇× u|−1/2∇× 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√(15)π/(5√(4455π−13568))≈11.2948 for the restricted constants. The unrestricted sharp constant remains open. As an application, bounded axisymmetric poloidal potentials admit no L2 zero mode of the Weyl-Dirac operator. An explicit conformal transformation extends this exclusion to both extremals at every real coupling.
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BibTeX
@misc{mederski2026critical,
title = {{Critical curl extremals and the hyperbolic plane}},
author = {Jarosław Mederski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41804},
eprint = {2610.41804},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41804},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Critical curl extremals and the hyperbolic plane
AU - Jarosław Mederski
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41804
UR - https://arxiv.org/abs/2610.41804
AB - We study the critical Sobolev-curl quotient on 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↦|∇× u|^−1/2∇× 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√(15)π/(5√(4455π−13568))≈11.2948 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.
ER -