Year: 2026
Type: article
Abstract: This paper studies the relationship between an analytic compactification of the moduli space of flat $\mathrm{SL}_{2}(\mathbb{C})$ SL 2 ( C ) connections on a closed, oriented 3-manifold $M$ M defined by Taubes, and the Morgan–Shalen compactification of the $\mathrm{SL}_{2}(\mathbb{C})$ SL 2 ( C ) character variety of the fundamental group of $M$ M . We exhibit an explicit correspondence between $\mathbb{Z}/2$ Z / 2 harmonic 1-forms, measured foliations, and equivariant harmonic maps to ℝ-trees, as initially proposed by Taubes. As an application, we prove that $\mathbb{Z}/2$ Z / 2 harmonic 1-forms exist on all reducible or Haken manifolds with respect to all Riemannian metrics. We also prove the existence of manifolds which support singular $\mathbb{Z}/2$ Z / 2 harmonic 1-forms but which have compact $\mathrm{SL}_{2}(\mathbb{C})$ SL 2 ( C ) character varieties, and this resolves a folklore conjecture.
Source: Inventiones mathematicae
Authors: Siqi He, Richard Wentworth, Boyu Zhang
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W7160617445