PreprintCC BY 4.0arXiv
Counterexamples to Grad's conjecture
Abstract
For each sufficiently large N, we construct smooth solutions of the magnetohydrostatic equations on embedded solid tori whose regular pressure levels are nested tori, whose magnetic field vanishes exactly on a round magnetic axis, and whose group of Euclidean symmetries, even when isometries reversing the sign of the field are admitted, is exactly the cyclic group CN of rotations through multiples of 2π/N. For each such N, these equilibria form a smooth one-parameter family which is locally nontrivial in the moduli space of embedded configurations and have none of the plane-reflection, axial, or helical symmetries conjectured by Grad. A Lean 4 certification of the above results is also provided.
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BibTeX
@misc{gomezserrano2026counterexamples,
title = {{Counterexamples to Grad's conjecture}},
author = {Javier Gómez-Serrano and Lukas Liehr and Mitchell A. Taylor},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24739},
eprint = {2609.24739},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24739},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Counterexamples to Grad's conjecture
AU - Javier Gómez-Serrano
AU - Lukas Liehr
AU - Mitchell A. Taylor
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24739
UR - https://arxiv.org/abs/2609.24739
AB - For each sufficiently large N, we construct smooth solutions of the magnetohydrostatic equations on embedded solid tori whose regular pressure levels are nested tori, whose magnetic field vanishes exactly on a round magnetic axis, and whose group of Euclidean symmetries, even when isometries reversing the sign of the field are admitted, is exactly the cyclic group C_N of rotations through multiples of 2π/N. For each such N, these equilibria form a smooth one-parameter family which is locally nontrivial in the moduli space of embedded configurations and have none of the plane-reflection, axial, or helical symmetries conjectured by Grad. A Lean 4 certification of the above results is also provided.
ER -