Non-optimal domains for the helicity maximisation problem

TL;DR

This work advances the helicity maximisation problem by deriving concrete geometric constraints on domains of fixed volume that would optimise the largest Biot–Savart eigenvalue $\lambda_+(\Omega)$. It develops a rigorous framework using curl eigenfields, harmonic Neumann fields, and a refined Hodge decomposition to relate interior spectral data to the geometry of solid tori and their tubular neighbourhoods, yielding explicit non-optimality criteria. The results include sharp constraints for axi-symmetric tori (non-optimal for aspect ratio $a\ge6$) and for tubular neighbourhoods of knots (rope-length-type bounds), as well as a general quantitative bound (involving $\Pi$, $\xi$, and $|\Omega|$) that precludes optimality under broad geometric conditions. These findings considerably narrow the class of potential optimisers, though the existence of any optimiser remains open and hinges on globally optimal geometric configurations under fixed volume and helicity constraints.

Abstract

In [J. Cantarella, D. DeTurck, H. Gluck and M. Teytel, J. Math. Phys. 41:5615 (2000)] the helicity isoperimetric problem which asks to find a smooth domain of fixed volume which maximises Biot-Savart helicity among all other smooth domains of fixed volume was initiated. It was shown that if an optimal domain exists, all of its boundary components must be tori. The present work extends these results by establishing additional geometric constraints which optimal domains, if they exist, must satisfy. This allows to rule out the optimality of a broad class of solid tori. The existence of optimal domains remains an open problem.

Theorems & Definitions (25)

• Theorem 1.1: CDGT002
• Theorem 1.3
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 3.1
• proof : Proof of \ref{['S3L1']}
• Corollary 3.2
• proof : Proof of \ref{['S3C2']}
• proof : Proof of \ref{['S1T3']}