Contact topology and intrinsic characterization of transversely vanishing curl eigenforms

Abstract

We study the geometry and topology of transversely vanishing eigenforms of the curl operator on $3$-manifolds, placing them in the broader framework of near-contact forms. We prove a Gray-type stability theorem for the kernels of near-contact forms, which we call near-contact structures, and derive a local neighborhood theorem around their singularities. As applications, we obtain a stability result for the contact topology of curl eigenforms, a classification of near-contact structures in terms of formal homotopical data, and a proof of the existence of contact structures on closed orientable 3-manifolds based on the spectral theory of the curl operator. We also study the symplectization of curl eigenforms as a possible bridge to symplectic dynamics. Finally, we characterize those transversely vanishing $1$-forms which are curl eigenforms for some smooth Riemannian metric. We deduce a generalization of the classical correspondence, due to Etnyre and Ghrist, between Reeb-like fields of contact forms and nowhere-vanishing rotational Beltrami fields to the setting of transversely vanishing vector fields.