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The study aims to explore the relationship between $ ext{SL}_{2}( ext{C})$ connections and $ ext{Z}/2$ harmonic 1-forms in closed 3-manifolds.

Z/2 harmonic 1-forms, ℝ-trees, and the Morgan-Shalen compactification

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The study aims to explore the relationship between $ ext{SL}_{2}( ext{C})$ connections and $ ext{Z}/2$ harmonic 1-forms in closed 3-manifolds.
Developed a correspondence between $ ext{Z}/2$ harmonic 1-forms, measured foliations, and equivariant harmonic maps to $ ext{R}$-trees.
Applied results to prove the existence of $ ext{Z}/2$ harmonic 1-forms on reducible and Haken manifolds.
Investigated manifolds with singular $ ext{Z}/2$ harmonic 1-forms and their compact $ ext{SL}_{2}( ext{C})$ character varieties.
$ ext{Z}/2$ harmonic 1-forms exist on all reducible or Haken manifolds regardless of the Riemannian metrics used.
Demonstrated the existence of manifolds supporting singular $ ext{Z}/2$ harmonic 1-forms with compact $ ext{SL}_{2}( ext{C})$ character varieties, resolving a folklore conjecture.

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Investigates the existence of Z/2 harmonic 1-forms in manifolds, highlighting important geometric implications.

He et al. (2026) studied this question.

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