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

TL;DR

This paper explores the connection between harmonic forms, measured foliations, and R-trees in the context of moduli space compactifications of flat connections on 3-manifolds, revealing new existence results and resolving a folklore conjecture.

Contribution

It establishes an explicit correspondence between harmonic forms, foliations, and R-trees, and demonstrates the existence of harmonic forms on all Haken manifolds, addressing a longstanding conjecture.

Findings

Existence of Z/2 harmonic 1-forms on all Haken manifolds.

Existence of manifolds with singular harmonic forms but compact character varieties.

Resolution of a folklore conjecture regarding harmonic forms and character varieties.

Abstract

This paper studies the relationship between an analytic compactification of the moduli space of flat $\mathrm{SL}_2(\mathbb{C})$ connections on a closed, oriented 3-manifold $M$ defined by Taubes, and the Morgan-Shalen compactification of the $\mathrm{SL}_2(\mathbb{C})$ character variety of the fundamental group of $M$. We exhibit an explicit correspondence between $\mathbb{Z}/2$ harmonic 1-forms, measured foliations, and equivariant harmonic maps to $\mathbb{R}$-trees, as initially proposed by Taubes. As an application, we prove that $\mathbb{Z}/2$ harmonic 1-forms exist on all Haken manifolds with respect to all Riemannian metrics. We also show that there exist manifolds that support singular $\mathbb{Z}/2$ harmonic 1-forms but have compact $\mathrm{SL}_2(\mathbb{C})$ character varieties, which resolves a folklore conjecture.