{"ID":23047525,"CreatedAt":"2026-09-17T05:47:22.378464935Z","UpdatedAt":"2026-09-17T05:47:22.378464935Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.18915","arxiv_id":"2609.18915","title":"Examples of $\\mathbb{Z}/2$-Harmonic 1-Forms","abstract":"In this paper, we develop methods for constructing $\\mathbb{Z}/2$-harmonic 1-forms in dimension three by varying the background metric. On $\\mathbb{R}^3$, we construct an example for which the complement of the smooth locus in the singular set is a Cantor set. We realize every finite graph with positive even valence at each vertex as the monodromy locus of a $\\mathbb{Z}/2$-harmonic 1-form on $B^3$. We also construct desingularization models for every critical $\\mathbb{Z}/2$-eigensection, agreeing exactly with the associated homogeneous model outside a compact set. Finally, we construct a nondegenerate $\\mathbb{Z}/2$-harmonic 1-form on every closed connected oriented three-manifold for a suitable smooth metric.","short_abstract":"In this paper, we develop methods for constructing $\\mathbb{Z}/2$-harmonic 1-forms in dimension three by varying the background metric. On $\\mathbb{R}^3$, we construct an example for which the complement of the smooth locus in the singular set is a Cantor set. We realize every finite graph with positive even valence at...","url_abs":"https://arxiv.org/abs/2609.18915","url_pdf":"https://arxiv.org/pdf/2609.18915v1","authors":"[\"Jiahuang Chen\",\"Siqi He\"]","published":"2026-09-16T16:55:33Z","proceeding":"math.DG","tasks":"[\"math.DG\"]","methods":"[]","has_code":false}