Rectifiability of divergence-free fields along invariant 2-tori

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