Contact topology and intrinsic characterization of transversely vanishing curl eigenforms

@misc{cardona2026contact,
  title = {{Contact topology and intrinsic characterization of transversely vanishing curl eigenforms}},
  author = {Robert Cardona and Willi Kepplinger},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36212},
  eprint = {2610.36212},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36212},
}

TY  - UNPB
TI  - Contact topology and intrinsic characterization of transversely vanishing curl eigenforms
AU  - Robert Cardona
AU  - Willi Kepplinger
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.36212
UR  - https://arxiv.org/abs/2610.36212
AB  - 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.
ER  -