Rectifiability of divergence-free fields along invariant 2-tori

Abstract

AbstractWe find conditions under which the restriction of a divergence-free vector field B to an invariant toroidal surface S is rectifiable; namely constant in a suitable global coordinate system. The main results are similar in conclusion to Arnold’s Structure Theorems but require weaker assumptions than the commutation $$[B,\nabla \times B] = 0$$ [ B , ∇ × B ] = 0 . Relaxing the need for a first integral of B (also known as a flux function), we assume the existence of a solution $$u: S \rightarrow {\mathbb {R}}$$ u : S → R to the cohomological equation $$B\vert _S(u) = \partial _n B$$ B | S ( u ) = ∂ n B on a toroidal surface S mutually invariant to B and $$\nabla \times B$$ ∇ × B . The right hand side $$\partial _n B$$ ∂ n B is a normal surface derivative available to vector fields tangent to S. In this situation, we show that the field B on S is either identically zero or nowhere zero with $$B\vert _S/\Vert B\Vert ^2 \vert _S$$ B | S / ‖ B ‖ 2 | S being rectifiable. We are calling the latter the semi-rectifiability of B (with proportionality $$\Vert B\Vert ^2 \vert _S$$ ‖ B ‖ 2 | S ). The nowhere zero property relies on Bers’ results in pseudo-analytic function theory about a generalised Laplace-Beltrami equation arising from Witten cohomology deformation. With the use of de Rham cohomology, we also point out a Diophantine integral condition where one can conclude that $$B\vert _S$$ B | S itself is rectifiable. The rectifiability and semi-rectifiability of $$B\vert _S$$ B | S is fundamental to the so-called magnetic coordinates, which are central to the theory of magnetically confined plasmas.